Tough situation. If I'm understanding you, you are talking about a long distance thing? How far is the trip? Why was it a couple of months between seeing her?
I'll answer your last question first. NEVER. Don't ask her to be your girlfriend. Just have her be your girlfriend. If she wants to ask, let her, and it's OK at that point to validate her by saying, "No, I'm just a one night stand gone awry.. kidding, of course you're my girlfriend". Otherwise, just be confident once you've attained that kind of relationship.
Now, back to your issue: if she is long distance, it's going to be an uphill battle. I've done it myself a few times, and it's led to long term relationships, so the good news is that you're speaking to a veteran
This is the thing-- she has to be really into you to make it work. HB9s get a lot of attention, so it won't be about "I want a boyfriend", it will be "I really like this guy at uni". Re: your question of how often you should be in touch, you need to give her just enough space that she can chase you, and that there is still mystery. Odds are that your friends are thinking about the end game. After you two are an item. In the interim, you need to be there or else you can't build a relationship, on the other hand you two can't exchange texts too regularly until more trust is built because we chase only that which doesn't chase us.
However, I should point out, that a relationship requires repeated interactions. Too many, too fast, however, and you'll kill the thing.
On the surface, it sounds like she likes you enough. The engagement ring thing sounds like a shit test, though, rather than a IOI. She is putting you on the spot to see how you react. The reflection thing is interesting. Could be she got carried away because you make her feel comfortable, or it could be that there's some third factor. You also need to be weary that just because a girl likes kissing you doesn't mean she wants to be your GF. Guys are sometimes just fillers. Guys who are there for a little bit, and it's nothing serious because otherwise a girl loses her confidence. Unfortunately, a guy you see every couple of months would fit the bill.
Keep in mind that too much comfort early on doesn't help you. If she is too comfortable with you, she won't work at it, and consumer behavior modeling says nothing is worthwhile if you don't have to work for it.
My suggestion is to reassess the situation. Decide if she likes you a lot, or if she is using you for confidence. If it's the former, take it slow, understanding there's a risk that someone else swoops in. In the meantime, keep seeing other girls.
Good luck. Let me know what happens.