Since the OP is also asking for personal understanding of 1 Cor 11 on head coverings, I will share my opinion. No one else has ever said what I think from what I have read, and even my wife disagrees with me. But this is how I see it.
There are a few things to consider first.
The early believers did not "go to church" as understood today. They met in homes. The Jewish believers met in synagogues at first, but were kicked out by Jews who rejected Christ. Greek believers, of which Corinth was comprised, never met in the pagan temples, but in homes. Jewish believers in Jerusalem met in the Temple until cast out, but also met in homes to break bread together (Acts 2:46). Lydia had invited Paul & company into her home (Acts 16:15), which was a common practice which is why the Didache gave practical guidelines for entertaining traveling evangelists and apostles. In fact, church as we know developed around the 3rd century AD when Greek orators became Christians and saw a way to make a living preaching and teaching. Paul, of course, did the same thing, as seen in Acts 20, but it was not necessarily a weekly meeting for him. The point is this: the Corinthians were meeting and eating in homes, which was an interactive gathering of all the believers instead of just one man preaching or teaching.
The other consideration is the context surrounding chapter 11. What were the main points Paul was addressing? A short survey of the chapters will give context to the head covering issue, which really seems out of place and a non sequitur passage.
The Corinthians were still carnal and natural in their thinking, in need of really renewing their minds. They operated in the spiritual gifts, but lacked spiritual maturity. A common sight today with ministers who have had miraculous ministries only to fall in scandal of immorality.
Perhaps they thought they were mature because they functioned in the gifts. A trap common to many. Here is the survey.
Church Division
C1 Church Division - they were taking sides on leaders
C2 Spiritual Discernment - Paul demonstrated Christ by his own self denial & the Spirit
C3 Division Caused by Natural Thinking - taking sides blinded them to group labor in the Lord
C4 Apostolic Foundation - Paul's self denial lowered their esteem of him, yet it allowed Christ to be seen
Natural Thinking and Carnal Behavior
C5 Natural Thinking Carnal Behavior - they thought they were spiritual tolerating sexual immorality
C6 Carnality in Lawsuits and Sexual Immorality - instead of turning the other cheek, they sued each other and visited prostitutes
C7 Marriage in the Lord - issue of sex meant for marriage, but not all marry, others married to unbelievers
Consideration for One Another When Coming Together
C8 Consideration for One Another - legalism brought judgment on those eating meat to idols
C9 Paul's Consideration for the Gospel by Declining Money - some leaders were seeking support, but viewed that as the low road whereas the high road is not to financially burden them
C10 Forsake Division & Carnality for Unity of the Spirit - graciously accepting food offered
Flowing in the Spirit when Together
C11 Flow of the Spirit and Unity in Fellowship - prayer & prophecy, communion meals
C12 Spiritual Gifts - distribution of spiritual gifts in the body
C13 Love as motive for Gifts - seeking gifts to bless others
C14 Prophecy and Tongues - share in an orderly fashion without confusion
The Future
C15 On Glorified Bodies - resurrected bodies will be a new spiritual body
C16 Mission Journeys - offering for poor Jewish saints in Jerusalem, his travels
I hope this is not too long. Here is the meat. As seen by the outline the Corinthians were a disorderly bunch and Paul tried to bring order to their gatherings.
Paul sets up the divine order in 11:3
1Co 11:3 But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
Natural thinking considers the word "head" in terms of authority. But the word "head" also means source, as in the head waters of a river is the source of the river. Source of what? The flow of divine life and communication.
God is the head of Christ; Christ is the head of man; man is the head of woman. I believe he means in marriage, not just any man or woman. In verse 4 he qualifies what he means.
1Co 11:4 Every man praying or prophesying, having his head covered, dishonoureth his head.
When a man covers his head he dishonors his head. What is a cover? Something that blocks or hides something else. What is the head of man? Verse 3 said, Christ. So what does 4 mean by context? It means when a man is talking to God or speaking on behalf of God, when he blocks or hides the communication of Christ he dishonors Christ. Now what about the wife?
1Co 11:5 But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven.
Her head is her husband. Her husband should be a spiritual leader for her. If he does not step up to the task, and she has to step in to pray or prophesize, it undermines his leadership and is a shame brought on him. To say it is as if she were shaven, I think he means as though she were not married.
Marriage of a woman throughout world history has always been a shame if they were not married. Today this seems very strange since so many women do not marry and there is no shame. I think this is what verse 6 is about.
1Co 11:6 For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered.
Change the word "covered" to married and "shorn" to single, then the verse is saying if she is not married, let her pray or prophesize freely as a single woman, since no husband is being undermined. But if she is ashamed of being single, let her marry.
The divine order does not change.
1Co 11:7 For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
1Co 11:8 For the man is not of the woman; but the woman of the man.
1Co 11:8 For the man is not of the woman; but the woman of the man.
Men should bring God glory by honoring Christ in prayer and prophecy, and wives should bring God glory by honoring their husbands. When the divine order is confused, it opens the door to problems.
1Co 11:10 For this cause ought the woman to have power on her head because of the angels.
I don't think these means holy angels, but demons. Confusion in order of the Lord is the play ground of devils where they reek all kinds of damage.
This order is not a power trip. It is part of mutual submission.
1Co 11:11 Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord.
1Co 11:12 For as the woman is of the man, even so is the man also by the woman; but all things of God.
As Paul continues this theme in chapter 12 when he speaks of part of the body have equality, though not sameness. God gives balance between seemingly more honorable roles, such as apostles with seemingly less honorable roles such as servants, by bringing more kingdom honor to the seemingly weaker vessel, ie servants, and here between husbands and wives.
Paul uses nature to make his point by comparing a wife praying instead of her husband (uncovered) with the embarrassment of nudity.
1Co 11:13 Judge in yourselves: is it comely that a woman pray unto God uncovered?
1Co 11:14 Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him?
1Co 11:15 But if a woman have long hair, it is a glory to her: for her hair is given her for a covering.
For a man to act like a woman it is a shame. Long hair on a woman covers her nudity, keeping her modest. But Paul, nonetheless, dismisses the entire argument suggesting the freedom we all have in Christ, but the considerations are discussed because we still live in a fallen world.
1Co 11:16 But if any man seem to be contentious, we have no such custom, neither the churches of God.
I suppose, most who took the time to read all of this will disagree. That is fine. My wife disagrees, too. I am just saying it for whoever this might benefit.