A bit of context
As a young adult who enjoys discourse and challenging the opinions of others, I frequently run into the problem of people hating me or my peers for challenging their opinions (primarily what they consider the proper course of things. Ie: the usage of their authority).
I'm not asking if it is acceptable to challenge their authority, merely the way they use it.
To reiterate my question in a more specific manner,
According to the bible, how much authority does the individual have in going to an authority figure to suggest a different method of the usage of their power, to call out a flaw in their teaching, or to call out an action as wrong, and how should one go about doing so?
Examples
1: Calling out an authority figure when they make a claim or command and expect you to agree (again, I'm not asking whether or not these things are in their power) merely because they have authority.
2: If something they taught is in direct opposition to the scripture.
3: If something seems to be in opposition to the scripture and asking for clarification, and/or using the Socratic method to come to a conclusion.
4: Requesting a better boundary to be employed that may not be contrary to scripture but have a better alternative. This can be rules set for children (Ie: no video games).
5: Requesting a better method to be employed that may not be contrary to scripture but have a better alternative. This can be how the authority figure uses their tone or words, or how they utilize their authority.
It all depends on the manner of how you and your peers are doing this and precisely what it is that is being addressed. The verses below address both sides.
Research
I'm not sure where to begin to find an answer for my question. I can think of a few verses that may help, but nothing that directly gives an answer.
1 Timothy 4:12 Let no one despise your youth, but be an example to the believers in word, in conduct, in love, in spirit, in faith, in purity. This seems unclear to me, but I can't tell whether letting no one despise your youth is active in the sense that I should call them out or if the setting an example is the answer to how to let no one despise your youth.
To add some context:
1Ti 1:3 As I urged you when I was going to Macedonia, remain at Ephesus so that you may charge certain persons not to teach any different doctrine, (ESV)
Given the context, then, we cannot extend 4:12 in a general sense to those who are younger. Timothy must have been somewhat young but put in a position of authority by Paul. Paul is telling him not to let anyone call his authority into question just because he was likely younger than most in such a position.
Proverbs 9:7 “He who corrects a scoffer gets shame for himself, And he who rebukes a wicked man only harms himself. 8 Do not correct a scoffer, lest he hate you; Rebuke a wise man, and he will love you. This merely states that if a man hates you for correcting him, he's a scoffer and a wicked man.
Beyond this, I don't have anything besides certain passages like Titus 2 that state what an authority figure's conduct should be like.
1Th 2:10 You are witnesses, and God also, how holy and righteous and blameless was our conduct toward you believers.
1Th 2:11 For you know how, like a father with his children,
1Th 2:12 we exhorted each one of you and encouraged you and charged you to walk in a manner worthy of God, who calls you into his own kingdom and glory.
1Ti 3:1 The saying is trustworthy: If anyone aspires to the office of overseer, he desires a noble task.
1Ti 3:2 Therefore an overseer must be above reproach, the husband of one wife, sober-minded, self-controlled, respectable, hospitable, able to teach,
1Ti 3:3 not a drunkard, not violent but gentle, not quarrelsome, not a lover of money.
1Ti 3:4 He must manage his own household well, with all dignity keeping his children submissive,
1Ti 3:5 for if someone does not know how to manage his own household, how will he care for God's church?
1Ti 3:6 He must not be a recent convert, or he may become puffed up with conceit and fall into the condemnation of the devil.
1Ti 3:7 Moreover, he must be well thought of by outsiders, so that he may not fall into disgrace, into a snare of the devil.
1Ti 5:1 Do not rebuke an older man but encourage him as you would a father, younger men as brothers,
1Ti 5:2 older women as mothers, younger women as sisters, in all purity.
...
1Ti 5:19 Do not admit a charge against an elder except on the evidence of two or three witnesses.
1Ti 5:20 As for those who persist in sin, rebuke them in the presence of all, so that the rest may stand in fear.
1Ti 5:21 In the presence of God and of Christ Jesus and of the elect angels I charge you to keep these rules without prejudging, doing nothing from partiality.
2Ti 2:15 Do your best to present yourself to God as one approved, a worker who has no need to be ashamed, rightly handling the word of truth.
2Ti 2:16 But avoid irreverent babble, for it will lead people into more and more ungodliness,
...
2Ti 2:22 So flee youthful passions and pursue righteousness, faith, love, and peace, along with those who call on the Lord from a pure heart.
2Ti 2:23 Have nothing to do with foolish, ignorant controversies; you know that they breed quarrels.
2Ti 2:24 And the Lord's servant must not be quarrelsome but kind to everyone, able to teach, patiently enduring evil,
2Ti 2:25 correcting his opponents with gentleness. God may perhaps grant them repentance leading to a knowledge of the truth,
2Ti 2:26 and they may come to their senses and escape from the snare of the devil, after being captured by him to do his will.
Tit 1:5 This is why I left you in Crete, so that you might put what remained into order, and appoint elders in every town as I directed you—
Tit 1:6 if anyone is above reproach, the husband of one wife, and his children are believers and not open to the charge of debauchery or insubordination.
Tit 1:7 For an overseer, as God's steward, must be above reproach. He must not be arrogant or quick-tempered or a drunkard or violent or greedy for gain,
Tit 1:8 but hospitable, a lover of good, self-controlled, upright, holy, and disciplined.
Tit 1:9 He must hold firm to the trustworthy word as taught, so that he may be able to give instruction in sound doctrine and also to rebuke those who contradict it.
1Pe 5:1 So I exhort the elders among you, as a fellow elder and a witness of the sufferings of Christ, as well as a partaker in the glory that is going to be revealed:
1Pe 5:2 shepherd the flock of God that is among you, exercising oversight, not under compulsion, but willingly, as God would have you; not for shameful gain, but eagerly;
1Pe 5:3 not domineering over those in your charge, but being examples to the flock.
Of course, it is a two-way street:
1Th 5:12 We ask you, brothers, to respect those who labor among you and are over you in the Lord and admonish you,
1Th 5:13 and to esteem them very highly in love because of their work. Be at peace among yourselves.
1Pe 5:5 Likewise, you who are younger, be subject to the elders. Clothe yourselves, all of you, with humility toward one another, for “God opposes the proud but gives grace to the humble.”
1Pe 5:6 Humble yourselves, therefore, under the mighty hand of God so that at the proper time he may exalt you,
(All ESV.)
There are also many other verses about the conduct of all Christians in general. Youth are to always show respect towards those who are older and all are to show respect to those whom God has placed in spiritual authority over them. There is conduct specific to overseers (elders) but there is also general conduct which applies to everyone.