(Note that while I marked this (thinking mainly about the first topic) with the Feminism tag, that is because it is relevant to feminism; I hold that it is perfectly compatible with (but not inspired) second-wave feminism but not compatible with third-wave feminism)
From the last point in that (long) cut, I want to point out some term usage (as I use them):
- Category-membership is an external classification based on someone's attributes. Not everyone's world-of-terms will agree as to how people-or-things should be categorised; everyone makes their own categories into which they put other people, and category systems neither do nor should agree on everything.
- Identity is a decision to make a self-categorisation part of a public statement as to what the flavour of one's person is. It often implies loyalty or accompanying cultural membership. Identification is self-built.