Well, this would be one reason why (anti-feminist snide remarks to the contrary) taking a course or reading a book or just talking to well-spoken knowledgeable people about feminist theory can be useful.
Feminism has gone farther than the equal rights movement, but not with the goal of having women be superior to men and become dominating.
However, often people who are sensitive to such issues may have a lot of anger, irony and/or sarcasm, and may be so used to their own thinking and language that they may tend to come across as they have seemed to you.
Specifically, I would guess that if you talked to a thoughtful feminist who said anything like what you heard, you would tend to hear them correct the impression they gave you, perhaps something like as follows:
“when I hear some feminists say that women can do anything a man can do and better”
-> May have intended to say that specific women could do practically anything better than specific men. Really the point being that sex alone should never be used to rule out that a man is going to be better at something than a woman. It may also have been collapsed with the observation that women often have perspectives and abilities that men often do not tend to have so much, so often women may be better than men at many tasks that are not traditionally considered so. It could also be collapsed with an affirmation of gender-fluidity.
“or that women should stick together against men and never talk bad about each other”
-> They might have meant, in certain contexts. For example, by “talk bad” they might mean in ways that involve common anti-feminine patterns such as slut-shaming or double-standards or others ready tools that can be used to put down one woman but that also put down all women.
“or that we should start companies and pay men less than women.”
-> Perhaps it was suggested as a pointed demonstration or “affirmative action” style idea. Not sure what they could’ve meant, but it wouldn’t tend to involve just reversing the target of discrimination except as a way to mirror how unbalanced existing behavior is.