Women and men can't have equal rights. For example if a man was to take sick leave it would go on a sick record, but if a women with HG was off sick then rightly so it wouldn't be classed as sick iygwim.
I don't understand this. 
Sure, men don't get HG. But they have the right to leave when they are incapacitated for medical reasons, don't they? Likewise, a woman who can't have children has the right to that leave. It's perfectly equal. You're (I think) conflating the right to have that leave, with the form the cause for needing it took. There's no need.
Btw, is HG leave not sickness leave? Purely an ignorant question - I'd assumed it would be medical leave of some form or other.
Anyway.
I prefer 'women's liberation' to 'feminism' when I'm being pedantic. It sounds very 70s but is easier to understand. Women need to be freed from the constraints placed on us by society.
'Equal rights' I'd understand as a general term, not always to do with women and men, but also applying to race and sexual orientation and so on. It just means you shouldn't be an arse for the sake of it, surely?
Sexism is discrimination based on sex. It is often (wrongly) conflated with misogyny, which IMO refers to a more structural form of discrimination against women, based on the (irrational) bias against women that is encoded in society. Sexism is to misogyny as racism is to white supremacy. So, you can find sexist women and racist people of colour. But, 'black supremacy' as a movement would be largely powerless, while 'white supremacy' is and has been hugely powerful, because it's able to exploit pre-existing biases in society. In the same way, misogyny is a form of oppression much more powerful than 'mere' sexism.