As you see, these are really good questions. I’ll let others deal with lofty issues, here are some observations.
First problem I see is that you feel it difficult to ask these questions. This is something that you should be able to talk about with your family/friends/school on a regular basis, not just in a “crisis mode” looking for excuses. So, think how to change your environment and relationship to achieve that.
One answer I heard from a Rav, who is a Cohen. A feminist lady approached him stating that “you can’t imagine how I feel when being on the other side of the mehitza”. He said – I can, when I was not able to attend the funeral of my beloved stepfather (as a Cohen). .. The point is that there are a lot of differences between different people and lots of mitzvos correspond to circumstances. I am still fuming that I still was not able to pick up a captured woman during a war.
Mitzvos that women have are way enough to fill your whole day doing – and learning how to do them well. Teaching children, making family a place of kedusha and many others will take years to learn how to do properly. In general, mitzvos bein adam l’havero are more complicated as they depend on the person – you talk differently to different people, so getting a PhD in psychology would be a good first step…
Re: men lording over women. Not sure what is going on in your house, you are welcome to stop by mine to disabuse yourself of this notion. Anyway, finishing writing, wife just called to throw the garbage away. Hinei I am accepting upon myself this mitzva of havata raeha kmoha.