A person trying to be a 君子 is not necessarily revealed to be a 伪君子，but the one who pretends to truly is, assuming I got your first question correctly.大家好！
I just learnt the word 伪君子 from a TV show. Is this word only used if the person is trying to be a 君子, or could it be for anyone? (Also, is this term only used for men? Is there a different term for women?)
In my opinion, 圣母 does not necessarily refer to a hypocrite. Usually 圣母s do believe in their values and they are not really prentending to behave that way.I wouldn't call a woman 伪君子，it's almost always men who try to be 君子。
But women can also pretend to be better than they really are, or being pretentious.
In these contexts I may say 这女的特虚伪，这女的特假，这女的特别（能）装，这女的特喜欢装好人，or using a new term 这女的特（喜欢装）圣母。
Edit: And Chinese women don't fancy the idea of being a 君子 either. In their minds women sometimes have a right not to be very moral because women are born to be at a disadvantage against men, so it's not right to require women to follow rules like men do. You may often hear women in a movie say something like, 我是女人，不是君子，or 我们女人才不当君子。
Of course this is more used in movies, as something to make the plot funny, not that women in reality all behave like that.