Ask her out on a friendly date.
I think I would bring the parents topic at one point when getting to know a guy I like, guys usually tend to like girls who look like their mothers. So it would be another test to see if we fit... I don't really care about it but it is something I would be curious about. I guess she's too.
As for the mixed feelings and back and forth feeling about the relationship, she is shy, some days she'd be more open up, others she'd be shy... same with mirroring. It is too soon for a relationship if she is still like that, give her time to feel really free around you. Maybe it will only come when you are alone only two of you and not in public. Anyways, give her time and spend some time with her, start with meeting her with her friends and later maybe only two of you.