Yes they do, but gender is more like a spectrum than it is a binary (or ternary) "choice". Its similar to sexual orientation in that sense. I know girls (or woman) that would rather build their own computer and write code than go shopping and I know boys (or men) that are the opposite. Its not that men and women are not different, its just that the stereotypes are wrong.
Yes, but if we prefer those things, we lie about it, because society wants us girls to be interested in certain things, or at least pretend to be. Guys love dumb girls. We don't have much choice but to pretend to be dumb if we want guys to be into us. So we reinforce that stereptype.