There's nothing wrong with setting some expectations with her for moving forward. It might help her become more comfortable asking questions of you and will help create boundaries that she can operate freely within.
Let her know that she won't upset/offend you if she asks similar questions of you.
Also keep in mind that many women are content to participate a conversation with a man but not lead/steer it...esp early on in a relationship.

start answering as if she asked you back just to see how she handles it

I never understand why people won't talk about themselves without questioning.

Give up. It's npt worth the time and energy

No, don't give up on her if things are going well.