Our distress and affliction - Hiebert notes that...

The two nouns, "distress" (anagke) and "persecution," (thlipsis) are linked together under one article, thus forming two aspects of the whole difficult situation in which they find themselves. The former denotes a necessity or compulsion that forces itself upon one, while the latter indicates the crushing pressure to which one is being subjected. Both terms refer to the difficult external circumstances confronting the missionaries. It is precarious to seek to identify their precise cause or character. Findlay points out that the former "signifies outward constraint, whether of circumstances or duty" while the latter points to "trouble from men." But the word rendered "in" is again the preposition epi, "over," and implies that the distress and afflictions continued but the missionaries were able to surmount them and no longer feel them as evil. This helped to make the report so welcome. This reference to their own distress and affliction is a tactful reminder to the readers that Paul was also facing difficulties while away from them. (Hiebert, D. Edmond: 1 & 2 Thessalonians: BMH Book. 1996)

Distress (318)(anagke {also transliterated as ananke} from ana = up, again, back, renewal, repetition, intensity, reversal + agkale = arm when bent) refers to any necessity or compulsion, outer or inner, brought on by a variety of circumstances. It can mean necessity imposed either by external conditions or by the law of duty.

Anagke - 17x in 17v (see below) - Matt 18:7; Luke 14:18; 21:23; Rom 13:5; 1 Cor 7:26, 37; 9:16; 2 Cor 6:4; 9:7; 12:10; 1 Thess 3:7; Philemon 1:14; Heb 7:12, 27; 9:16, 23; Jude 1:3.  NAS = compulsion(3), constraint(1), distress(3), distresses(1), hardships(1), inevitable(1), necessary(2), necessity(3), need(2), obliged*(1).

The idea of  trouble, distress or hardship is dominant in Luke 21:23; 1Co 7:26; 2Co 6:4; 2Co 12:10. This is the predominant sense here in 1Thessalonians 3:7 where anagke refers to difficult circumstances that come on one with compelling force.

The idea of an obligation of compelling nature (a complete obligation, a necessary thing) is dominant in  Mt 18:7; Lk 14:18; Ro 13:5; 1Co 7:37; 9:16; 2Co 9:7; Philemon 14; Heb 7:12, 27; 9:16, 23; Jude 3; Lk 23:17. Anagke can be subdivided into a moral necessity (as in Mt 18:7; Heb 7:12, 27; 9:16, 23) or a spiritual necessity (Ro 13:5; 1Cor 9:16; Jude 1:3)

Anagke refers to constraint (which is the act of being checked, restricted or compelled either to avoid or to perform some action) which is either a result of external pressures (compelling forces) or moral sense of duty.

It refers to that which is necessary due to the pressure of circumstances and so refers to that which is inevitable.

It refers to a necessity (an indispensable thing,  something needed for a desired result) or constraint as inherent in the nature of things.

Anagke is used of outward calamities or distresses as in Luke 21:23.

Anagke is used 21 times in the Septuagint (LXX) (1Sa 22:2; Esther 4:17; Job 5:19; 7:11; 15:24; 18:14; 20:22; 27:9; 30:25; 36:19; Ps. 25:17; 31:7; 107:6, 13, 19, 28; 119:143; Prov. 17:17; Jer. 9:15; 15:4; Zeph. 1:15). Here are a few representative uses...

Job 27:9 "Will God hear his cry, when distress (Lxx = anagke) comes upon him?

Psalm 25:17 The troubles of my heart are enlarged. Bring me out of my distresses (Lxx = anagke).

Psalm 31:7 I will rejoice and be glad in Thy lovingkindness, Because Thou hast seen my affliction; Thou hast known the troubles (Lxx = anagke) of my soul,

Psalm 107:6 Then they cried out to the LORD in their trouble; He delivered them out of their distresses (Lxx = anagke). (same idea in Ps 107:13, 19, 28)

Psalm 119:143 Trouble (Lxx = thlipsis) and anguish (Lxx = anagke) have come upon me; Yet Thy commandments are my delight.

Zephaniah 1:15 A day of wrath is that day, A day of trouble (Lxx = thlipsis) and distress (Lxx = anagke), A day of destruction and desolation, A day of darkness and gloom, A day of clouds and thick darkness (Refers to Day of the Lord)

NIDNTT has an interesting background note on the Greek ideas behind the words for necessity, must and obligation...

The Greek approach to life was largely shaped by the consciousness of a necessity in existence and events which is subject to the laws and norms of fate. The impersonal verb-form dei (it is necessary) is the most comprehensive expression for this life. It reflects the sense of a determining constraint, no matter whether it was exerted by magic or laws, by men or by gods.

The general concept which expresses this feeling of having been consigned to fate (a situation which not only embraces human existence, but, in personifying a universal principle, also dominates the lives of the gods themselves) is ananke (anagke).

By contrast with the words which are associated with fate and nature the verb opheilo originally belonged to the legal sphere; it expressed initially one’s legal and economic, and then later one’s moral, duties and responsibilities to the gods and to men, or to their sacrosanct regulations. Whereas opheilo also expresses human and ethical responsibility in the NT, the typically Greek concepts of dei and ananke could only be taken over with some modification which personalized and re-interpreted them so that they express the will of God.

All words based on the word-stem anank- (anagk-) denote in varying gradations every form of outward or inward pressure which is exerted on men.

For the Greeks ananke (anagke) was the power which determined all reality, the principle which dominated the universe. At various times men ascribed a divine character to it; Plato (Leg. 818c) in fact ranked it higher than the gods. Man is under a constraint because of his natural being; the final limitation of his existence by death is also part of this compulsion (cf. W. Grundmann, TDNT I 344 f.).

In the OT the naturalist outlook of the Gk. world is replaced by a historical one. ananke translates several Heb. words which denote the afflictions and distresses of illness, persecution, enmity etc. which were often taken by Israelites to indicate God’s alienation from them...

 In the last resort, it is Yahweh alone who can save men from ananke (Ps 25:17), and even lead a person into it (Job 20:22; cf. Jer 9:14; 15:4). He will raise up the great ananke on the day of his wrath (Zeph 1:15), a conception which had powerful effect on the thought of post-exilic Judaism. (Brown, Colin, Editor. New International Dictionary of NT Theology. 1986. Zondervan) 

Richards writes that anagke refers to an...

inner compulsion. Words in this group indicate inner or outward pressures. In some cases the words simply refer to troubles or afflictions--pressures that cause suffering. In most cases, however, they are linked with the channeling of choices or the constraints that cause one to choose. In the NT, outward force is not primarily intended. One is not forced to choose against his will, but the forces suggested act on and with the will. (Richards, L O: Expository Dictionary of Bible Words: Regency)

Vincent writes that anagke

 means originally force, constraint, necessity, and this is its usual meaning in classical Greek; though in the poets it sometimes has the meaning of distress, anguish, which is very common in Hellenistic Greek. Thus Sophocles, of the approach of the crippled Philoctetes: “There falls on my ears the sound of one who creeps slow and painfully (anagke)" and again, of the same: “Stumbling he cries for pain (anagke)"...The distress is that which should precede Christ’s second coming, and which was predicted by the Lord himself, Matt. 24:8

Rienecker writes that anagke means....

necessity, compulsion of any kind, distress, calamity. It refers to the whole state of things between the first and second coming of Christ (Godet). (Rogers, C L - originally by Fritz Rienecker: New Linguistic and Exegetical Key to the Greek New Testament. Zondervan. 1998)

Barclay writes that anagke...

literally means the necessities of life. Certain burdens a man may escape, but others are inescapable. There are certain things which a man must bear. The greatest of these are sorrow, for only the life which has never known love will never know that, and death which is the lot of every man. The triumphant endurance enables a man to face all that is involved in being a man. (Barclay, W: The Daily Study Bible Series. The Westminster Press or Logos)

Anagke is used 17 times in the NT...

Matthew 18:7 "Woe to the world because of its stumbling blocks! For it is inevitable (anagke) that stumbling blocks come; but woe to that man through whom the stumbling block comes! (Comment: Here anagke speaks of an inevitability or what is bound to be or to have to be. It is incapable of being avoided or evaded. In this verse it is expected that those in the world will cause {it is inevitable} Christians to be offended, stumble and sin, but they will be held responsible for causing them to stumble and they will be judged for it)

Luke 14:18 "But they all alike began to make excuses. The first one said to him, 'I have bought a piece of land and I need (echo = have + anagke = need) (I am constrained, "I am obliged") to go out and look at it; please consider me excused.' (It could be rendered "I have necessity" and as such is a strong expression).

Luke 21:23 "Woe to those who are with child and to those who nurse babes in those days; for there will be great distress (calamity) upon the land, and wrath to this people,

Romans 13:5 (note) Wherefore it is necessary to be in subjection, not only because of wrath, but also for conscience ' sake. (Necessity, imposed either by external conditions or by the law of duty. Denney explains, “There is a two-fold necessity for submission—an external one, in the wrath of God which comes on resistance; an internal one, in conscience. Even apart from the consequences of disobedience, conscience recognizes the divine right and function of the authority and freely submits to it.")

1Corinthians 7:26 I think then that this is good in view of the present distress,  that it is good for a man to remain as he is (Anagke  signifies here that which arises from the pressure of external circumstances, as a result of which people tend to do what they would not under other conditions)

 1Corinthians 7:37 But he who stands firm in his heart, being under no constraint, but has authority over his own will, and has decided this in his own heart, to keep his own virgin daughter, he will do well.

1 Corinthians 9:16 For if I preach the gospel, I have nothing to boast of, for I am under compulsion; for woe is me if I do not preach the gospel. (Paul is under a divine constraint which he cannot escape.)

2 Corinthians 6:4 but in everything commending ourselves as servants of God, in much endurance, in afflictions, in hardships, in distresses (Here anagke is used in the sense of suffering).

2 Corinthians 9:7 Let each one do just as he has purposed in his heart; not grudgingly or under compulsion (anagke); for God loves a cheerful giver. (Compulsion in English describes  an irresistible urge to behave in a certain way. In context anagke refers to external pressure and coercion, quite possibly accompanied by legalism. Believers are not to give based on the demands of others, or according to any arbitrary standards or set amounts). (Baker writes that "The word “compulsion” {anagke} does not carry the emotional baggage of “reluctantly.” It simply describes a difficult situation one is forced to endure. - 2Corinthians. The College Press NIV Commentary. page 330. College Press Pub)

2 Corinthians 12:10 Therefore I am well content with weaknesses, with insults, with distresses, with persecutions, with difficulties, for Christ's sake; for when I am weak, then I am strong. (One source states that anagke also "leans toward bodily violence, torture, and suffering")

1Thessalonians 3:7 (note) for this reason, brethren, in all our distress and affliction we were comforted about you through your faith;

Philemon 1:14 but without your consent I did not want to do anything, that your goodness should not be as it were by compulsion (pressure, necessity), but of your own free will. (Paul wanted Onesimus to minister alongside him, but only if Philemon openly and gladly agreed to release him. In other words, Paul was not trying to pressure or coerce him to do so.)

Hebrews 7:12 (note) For when the priesthood is changed, of necessity there takes place a change of law also (He is saying that the fact that the priesthood has been changed forces the conclusion that by necessity the entire legal structure on which the priesthood was based has been changed.)...7:27 (Jesus the better High Priest) Who does not need daily (the Levitical priests were compelled by the Mosaic law and the fact that they were sinners unlike the sinless Christ) , like those high priests, to offer up sacrifices, first for His own sins, and then for the sins of the people, because this He did once for all when He offered up Himself.

Hebrews 9:16 (note)  For where a covenant is, there must of necessity be the death of the one who made it (the provisions of a will are only promises until the one who wrote the will dies, this death being necessary to activate the promises)... 9:23 Therefore it was necessary for the copies of the things in the heavens to be cleansed with these, but the heavenly things themselves with better sacrifices than these (In other words, it was necessary for the copies to have sacrifices and it was therefore necessary for the better covenant to have better sacrifices.)

Jude 1:3 Beloved, while I was making every effort to write you about our common salvation, I felt the necessity to write to you appealing that you contend earnestly for the faith which was once for all delivered to the saints.

Affliction (2347)(thlipsis [word study] from thlibo = to crush, press together, squash, hem in, compress, squeeze in turn derived from thláo = to break) originally expressed sheer, physical pressure on a man. Medically thlipsis was used of the pulse (pressure). It is a pressing together as of grapes. It conveys the idea of being squeezed or placed under pressure or crushed beneath a weight. When, according to the ancient law of England, those who willfully refused to plead guilty, had heavy weights placed on their breasts, and were pressed and crushed to death, this was literally thlipsis. The iron cage was stenochoria. Thlipsis thus refers not to mild discomfort but to great difficulty.

Thlipsis - 45x in 43v - Matt 13:21; 24:9, 21, 29; Mark 4:17; 13:19, 24; John 16:21, 33; Acts 7:10f; 11:19; 14:22; 20:23; Rom 2:9; 5:3; 8:35; 12:12; 1 Cor 7:28; 2 Cor 1:4, 8; 2:4; 4:17; 6:4; 7:4; 8:2, 13; Eph 3:13; Phil 1:17; 4:14; Col 1:24; 1 Thess 1:6; 3:3, 7; 2 Thess 1:4, 6; Heb 10:33; Jas 1:27; Rev 1:9; 2:9f, 22; 7:14. NAS = affliction(14), afflictions(6), anguish(1), distress(2), persecution(1), tribulation(16), tribulations(4), trouble(1).

See related discussion of affliction in (1Th 3:3-note) and the verb suffer affliction in (1Th 3:4-note)

Philemon 1:7 For I have come to have much joy and comfort in your love, because the hearts of the saints have been refreshed through you, brother.

Application: How do you feel when you receive news of a brother or sister standing fast in their faith? stumbling in their faith? You are either encouraged or discouraged, respectively. And if you are going through distress and affliction yourself, such news can have either a positive or negative impact on your walk! Our Christian walk is seen and/or heard about by other believers, to whom we can be either a source of encouragement or discouragement - which effect characterizes your walk beloved of God?

Faith (4102)(pistis) here refers to their continuing state of reliance on Christ, walking by faith. The idea of their ongoing faith is similar to that conveyed by Paul in Colossians

As you therefore have received Christ Jesus the Lord (how? by faith), so walk in Him (how? by faith) (see note Colossians 2:6) (Ryrie comments: "Just as Christ is received by faith, the believer is also to walk (live) by faith, acknowledging the lordship of Christ over his life 2 Cor. 5:7)

Wayne Grudem defines faith that saves one's soul...

FOR NOW WE REALLY LIVE, IF YOU STAND FIRM IN THE LORD:  hoti nun zomen (1PPAI) ean humeis stekete (2PPAI) en kurio: (1Samuel 25:6; Psalms 30:5; Philippians 1:21) (John 8:31; 15:4,7; Acts 11:23; 1Corinthians 15:58; 16:13; Galatians 5:1; Ephesians 3:17; Ephesians 4:15,16; 6:13,14; Philippians 1:27; 4:1; Colossians 1:23; Hebrews 3:14; 4:14; 10:23; 1Peter 5:10; 2Peter 3:17; Revelation 3:3,11)

Live (2198)(zao) means to enjoy real life or have true life and worthy of the name Christian. The present tense speaks of Paul's desire that this be a continual experience. He really lives in comparison with his feelings of dread at not knowing how they were doing as the tempter was tempting them in the context of their appointed afflictions.

UBS Handbook explains live writing that...

The most likely explanation is that for Paul, as for Hebrew thought in general, life is not simply the opposite of death, just as peace is not simply the absence of war. There are degrees of being alive. Isolation from the community, like illness, can makes a person less alive.

In many languages, such an expression is not understood as mere figurative speech or interpreted as “spiritual truth,” but as reality. Conversely, the renewal of a relationship, like the restoration of health, increases life and makes it full. So it was with the renewal, though Timothy, of Paul’s relationship with the Thessalonians. Knox conveys this concept well: “it brings fresh life to us”; and so does Barclay: “it makes life worth living for us.” Paul’s life is bound up with that of the Christian communities he has helped to found.

Vine has an interesting note on now we really live explaining that it means to...

enjoy life; a vivid rhetorical description of the contrast between his state of apprehension when he was in ignorance of their welfare, and the comfort and ease of mind produced by Timothy’s good news. (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson or Logos)

Barnes adds that...

The same thing here affirmed is true of all faithful ministers of the gospel. They feel that they have something that may be called life, and that is worth living for, when those to whom they preach maintain a close walk with God. (Albert Barnes. Barnes NT Commentary)

EBC writes that ...

Paul had been given a new lease on life. To know they continued "standing firm in the Lord," unmoved by affliction and unshaken by his detractors, was enough to stimulate Paul to renewed activity.  (Gaebelein, F, Editor: Expositor's Bible Commentary 6-Volume New Testament. Zondervan Publishing)

Hiebert explains it this way noting that...

Before there had been a dead weight of apprehension; they felt lifeless and had no enthusiasm. But now in consequence of the news Timothy brought, we live feel like we truly live, not just exist. The present tense indicates that this is not just a momentary reaction but an abiding inspiration.

Paul's words are a vivid rhetorical description of the contrast between the former state of apprehension, when they were ignorant concerning the reaction of the Thessalonians under persecution, and the ease of mind and encouragement produced by the news Timothy brought. They felt that they had been given a new lease on life, could again go on living with a sense of fullness of power and satisfaction. Had the Thessalonians apostatized, it would have been a veritable deathblow to Paul. "The success of the cause of Christ, i.e., the winning of souls to Him, and the continuance of such souls in Him, was St. Paul's very life." The fact that their faith had proved itself steadfast was a demonstration that their work of proclaiming the gospel at Thessalonica had been successful, and this made life worthwhile for them. (Hiebert, D. Edmond: 1 & 2 Thessalonians: BMH Book. 1996)

Spurgeon wrote that...

Never is the servant of God so full of delight as when he sees that the Holy Spirit is visiting his hearers, making them to know the Lord, and confirming them in that heavenly knowledge. On the other hand, if God does not bless the word of his servants it is like death to them. To be preaching and to have no blessing makes them heavy of heart: the chariot-wheels are taken off, and they drag heavily along: they seem to have no power nor liberty.

If - (condition of the first class with present tense, active voice, indicative mood)  is not intended to question their stability and thus the NIV is accurate rendering it "since you are standing firm". However this sentence in addition to expressing his assurance that they are standing firm but also contains a reminder that they must continue to stand firm in the future.

Hiebert notes that although they are indeed at present standing firm,

the conditional form of the statement serves to remind them that they have a continued responsibility to "stand fast." The verb (steko) is a late formation, formed from the perfect tense of the verb histemi, "to stand," and carries the idea of firmness and stability. Here the present tense stresses that for the future they must go on standing firm, like soldiers repelling an enemy attack. In the Lord points out the sphere of their steadfastness. Their faith has brought them into a close vital union with "the Lord," Jesus Christ, and they must adhere steadfastly to Him.  (Ibid)

Stand firm (4739) (steko [word study] is corrupted from ésteka, perfect tense of hístemi = to stand) can mean to stand literally ("whenever you stand praying..." Mk 11:25) but is used figuratively in this passage meaning to stand firm in faith and duty, to be constant, to persevere, to remain steadfast, to continue in a state.

The present tense expresses Paul's desire that his spiritual children continue remaining firmly committed in their Christian belief. Remember that belief always demonstrates itself to be genuine by one's conduct or lifestyle.

Steko - 11x in 11v - Mark 3:31; 11:25; John 8:44; Rom 14:4; 1 Cor 16:13; Gal 5:1; Phil 1:27; 4:1; 1 Thess 3:8; 2 Thess 2:15; Rev 12:4. NAS = stand(2), stand firm(4), standing(1), standing firm(2), stands(2).

The word steko pictures an army that refuses to retreat even though it is being assaulted by the enemy. Standing firm as a believer is clearly an important exhortation as shown by Paul's frequent use illustrated in the following passages.

In 1Corinthians Paul later exhorted the believers to

Be on the alert, stand firm  (steko) in the faith, act like men, be strong. (1Cor 16:13) (Comment: Note that all four verbs are present imperative calling for this to be the believer's lifestyle or continual practice.)

In the Lord - In union with Him as a branch is in vital relation to the Vine. In the Lord is always and only the position and Person in Whom believers can be enabled to stand firm when the winds of affliction and the fiery darts of the tempter fly loose and fast all about us.

Lord (2962) (kurios) signifies sovereign power and absolute authority. It is the one who has absolute ownership and uncontested power.