A quick point, related to what I've been blogging about (understanding context of the times).
Whenever you read the word "judge" in the Bible, remember there are two totally different meanings, do not assume the first meaning, while is a legal judging, such as a magistrate might do. Yes, it is sometimes used in that context, but just as often it is used to mean:
a hero or military leader who is called by God.
That is why the Book of Judges is called, um, the Book of Judges, as it's about military leaders and heroes who led Israel when Israel did not yet have kings. These were not legal folks who rendered magistrate justice.
So while St Paul is indeed writing about legal matters and the need for justice and righteousness in 1 Cor 6, you have to remember that he, a zealous Jew (so zealous he turned Christians over for death before his conversion) would always have the two definitions of "judge" in this mind, somewhat intertwined. When he mentions "judging angels" he is speaking of the other definition, embedded within his discussion of more legal matters and definition.
He's not wrong to do this, but it is, as all epistles are, a bit of unsupported statement since producing long written documents was tough enough and Paul like others assumed the recipient of their letters understood many of the extenuating meanings and nuances that others would not outside of a previously Jewish context. In other words he was not writing in a way to explain to people who would later (like moderns) think "judge" always and only means magisterial.
Remember that Israel was an entity completely under and of God, not an organized government nor a kingship in its early days. Instead God chose people to lead, starting with Moses. When Josue (Joshua) died the people asked God who they should select as their next military leader. This is because the Israelites were still, through military force, claiming their promised land, so their leader's first and foremost job was to be a military leader, one that God approved and supported. It had nothing to do with being a judge of legal matters, so there were twelve, for lack of a better word, heroes of Israel whose job was to lead the Israelites in times of danger and/or in military matters. They were given the title of "judges," using, obviously, the second definition.
Thus it is very easy for Paul in both his thinking and his writing to slip from one definition of judge into the other. He is building an argument, in 1 Cor 6, that people need to be just in all matters and not be hypocrites, and he uses public litigation as the example that he gives, where he questions and chastises how people can be just if they are so unjust and be litigation crazy (yes they were already suing each other right and left to get money and settle scores in those times). In 1 Cor 6:7 he implies that people should not sue each other and just suffer even obvious wrongs in silence and without counter suit and legal remedies (turn the other cheek, as Jesus taught). He then, having raised the problem of legal fights and injustice, and the related problem of being unworthy to judge even small matters of life, say nothing of the large ones, in 1 Cor 6:1-8 onward, then shifts in 1 Cor 6:9 onward to discussing the unjust, unsanctified and unjustified sinners and their impending judgment from God in general.
So you see what I mean? He is writing about a problem he sees 1) being too litigious and also being unforgiving and unjust, 2) points out the flaws in that mindset, 3) introduces a higher more Christian mindset and 4) concludes with warnings what to avoid and 5) the sacredness of the body in service to Christ. Paul is starting with a specific problem in order to raise the level of thinking, action and sanctity of those he is writing to and 1 Cor 6 progresses accordingly.
So, how to understand 1 Cor 6:3 "Do you not know that we shall judge angels?" Even though that is smack dab in the middle of him speaking of earthly legal matters, he does not mean that humans will judge (as in adjudicate or assess) the actions of angels or the angels themselves. He is hearkening back to the second definition of judges, the heroism and God given leadership, that humans who are truly within Jesus Christ will receive from God. In an effort to get people's heads out of the petty earthly legalities Paul drags angels into his writing in order to uplift people to their correct perspective which is to be genuine leaders authorized by God through Christ, and it is then that yes, people have the right to "judge" as in to "lead into spiritual battle" angels.
That is not so far fetched. Do you not all each have a guardian angel that God gave you to accompany you always while you live? Most times do you even remember that your guardian angels exists? Do you lead your guardian angel in times of danger such as witnessing for Christ? Or do you forget you have a guardian angel?
Your actions on earth in your individual sanctified service to Christ is your training wheels, your practice, in leading ("judging") an angel.
When you are a saint (as in dead and gone to heaven), then you will, God willing, have a role in leading (as in "judging") angels on their home turf, heaven, regarding earthly matters.
It's not too tough to understand that those who did not partner with their guardian angel, with humility, while alive on earth aren't exactly leading the charge of the angel brigade in heaven when it is that time.
1 Cor 6:2 Do you not know that the saints will judge the world? And if the world will be judged by you, are you unworthy to judge the smallest matters?
So you see, Paul is basically scolding most severely the people who look forward to being saints and angel partners as being incapable of even keeping themselves out of greedy and vengeful legal lawsuits, and he is warning that they are not even on the first rung of "judging angels."
o Current situation, constant litigation about earthly matters such as money, power and revenge.
o The current situation must be improved by bringing the matters before the saints, which means in other words having a Christian response (such as turning the other cheek). Paul means the saints are (second definition) providing the example, the leadership of behavior. He means that just as the saints did not seek money, power and revenge through the courts, likewise people must turn to the saints for their example to shun that response likewise.
o In the future the saints (most of whom were martyrs) will judge the world because they, in heaven, will bring their petitions of injustice received before God.
And with the prayers of the saints there went up before God from the angel's hand the smoke of the incense. And the angel took the censer and filled it with the fire of the altar and threw it down upon the earth, and there were peals of thunder, rumblings, and flashes of lightning and an earthquake.
See, Paul wrote Corinthians well before John saw the revelation of the Apocalypse and in fact Paul was martyred decades before John died a natural death (the only one of the Apostles not to be martyred). Paul was dead for decades before John witnessed to Revelation, to the Apocalypse. Yet Paul correctly understood within the spirit of Jesus Christ that the sufferings of the martyrs and saints will accumulate before God's throne, and will in turn be justification for the End of Times. This is why the angel throws the censer with fire from the altar and the prayers of the saints onto earth to smite it.
And so now you can see that through the Holy Spirit the genuine Apostles and disciples of Christ are all reading from the same page, to use an old saying. Paul knew in his heart what John would later see in person, which is the role of the saints in judging (both meanings) the people of the earth. All who learned first hand from Christ understand the inevitability if not the timing of God's judgment of those alive on earth with the saints as witness and examples.
Now, the angels have never been and never will be part of earthly life, since angels abide within God in heaven and are not of time, matter or energy. Thus they of course are not "judged" by humans in a legal or a behavior sense. They are, however, available to be led by genuinely God appointed humans, but only after death. How do we know this? Because Jesus Christ himself never called upon an angel (though one was sent to him in the Garden of Gethsemane and others after the temptation). Jesus pointed out he could have called angels to protect him and destroy the city at any time, but he did not because that was not his purpose. Likewise the Apostles understood that the only angels on earth are the guardian angels, and that the combat will occur only at the End of Times.
So in a few sentences Paul is cramming together the mundane but very dire state of the litigious supposed Christians who are suing each other and pagans for money, power and revenge, the need for a higher thinking, and then, of course, the highest calling which will mostly occur in heaven, when the righteous and justified people are indeed among the angels.
This is a complicated but essential part of my point that one cannot understand the Bible if one does not understand the mental, spiritual and linguistic context of the various authors, all of whom write under the guidance of the Holy Spirit but in the language and meaning of that contemporary time.
I hope that this was helpful!