Why does he dishonour his head by covering it? Because in that age, in Greece, in Corinth, at that time, to cover the head meant you were a woman. The message of covering the head was that you were the supporter in marriage and in society.
Many congregations have laid no obligation on ladies to wear a head covering for generations, in common with the vast majority of Christian churches in the Western world. It is a matter of your private judgement, if you wish to wear a head covering or not. Years ago, in our churches, all ladies wore head coverings and they were important, because it was the culture and they meant something, but those head coverings were not what the apostle Paul is talking about when he writes to the Corinthians. He is talking about veils, which covered all the head and the face.
‘Every man praying or prophesying.’ This is about what is happening in church, of course, not just about what those leading the service are doing. Every man in this congregation, in the service, is praying and prophesying.