If you'e that sure that she likes you I don't see what's stopping you. Family wise with the religion issue, I wouldn't worry about - you're hardly marrying her just yet. The uni issue, sure you only have three months, but look at it the other way - you have three whole months that you could potentially spend together, and who says relationships have to end with uni?
I'd go for it, you'll only regret it if you pass up the opportunity.