Well, what religion is she (or is she nonreligious)? It might be helpful to know that, since it might help gauge how open she'd be to your faith. If half her posts on Facebook are quotes from Dawkins or Hitchens, you and her probably aren't going to get along.
I don't know yet. It's a very new thing, so we haven't had time to chat about that kind of thing yet. So far, we've been meeting before and after class. If she uses facebook, it's not with her real name (doesn't show up in the search). This week I'm inviting her to a post-liturgy brunch (and liturgy if she's interested) date, which is held in the ballroom in walking distance from the church.