It might be clearer to ask about Jewish versus other men and women. My impression is that most women wore head coverings at that time, and most men wore beards, but customs varied greatly among non-Jewish people. Prostitutes likely would not wear head coverings. I don't see homosexuals as a group that would be likely to have a common practice about beards, though doubtless some shaved to be more feminine.
Michael Marlowe discusses head coverings and other clothing issues at http://www.bible-researcher.com/headcoverings3.html . The Bible makes clear that the Jews are to remain a distinct people, and should not adopt many of the customs of the people around them (gentiles).
After the doings of the land of Egypt, wherein ye dwelt, shall ye not do: and after the doings of the land of Canaan, whither I bring you, shall ye not do: neither shall ye walk in their ordinances. (Leviticus 18:3. cf. also Deuteronomy 12:29-32 and 2 Kings 17:13-15.)
The LORD has prepared a sacrifice and consecrated his guests ... and on the day of the LORD's sacrifice, I will punish ... all who array themselves in foreign attire. (Zephaniah 1:8)
Marlowe suggests that Jewish men of the first century did not cover their heads when they prayed. They likely covered or uncovered their heads for practical reasons only. Women normally kept their heads covered any time they were outside of their home.
Philo of Alexandria refers to a woman's head covering as a "symbol of modesty", suggesting that an innocent woman would be uncomfortable in its absense.