I posted about head coverings before (I will add a link to my previous post if I find it)
To directly address, head coverings: they are not required. But I suppose a parish might have a dress code… :shrug:
From a historical point of view, head coverings where not required by Cannon Law until I believe 1919. Before that, women always wore them based on tradition.
Head coverings for women go back to the times before Christ. Women of Conservative and Orthodox Jewish synagogues still cover their heads today. But even the Roman Pagans covered their heads.
It originated as a cultural sign of modesty and chastity. Until World War I, long hair was considered a sign of virginity. While short hair was considered a lack of chastity.
For example: as part of the punishment of prostitution, the woman’s hair would be cut short.
Head covering served two purposes, to hid the length of the hair while veiling server to make the hair appear longer. Both were seen as signs of modesty.
By the end of World War I (perhaps because of the war itself) veiling was failing out of practice. So the Church issues a Cannon law to obligate the head coverings. Then by 60s and 70s, fewer and fewer women were using head coverings (almost none) so the Church removed the Cannon Law in 1983.
Even the St Paul mentions it in his letters, it was never really a subject of religion or theology. It was a cultural sign of modestly and Chastity for over 2100 years. St Paul’s reason for addressing it was to tell women that they should be dressing modestly in the House of God (which is something that we have a major issue with today).
I hope this makes sense and helps.