1) Depends entirely on your and her situation. If you really want her in particular keeping at it for months or even years might be worth it, but it also depends on how serious and long term her issues are. The problem is that there's a chance you'll waste a lot of time on her and miss out on other opportunities. It sounds like you're in a quasi-relationship now, so I'd let that develop for a while, assess how you feel about the situation, and re-check her feelings on the matter.
2a) No, you should still try to move it forward, just make sure to give her some extra room and that she shows interest in building or maintaining your relationship as well.
2b) I'm sure they exist, but in my experience are extremely rare.
3) Your guess is probably better than mine.
Your situation reminds me a bit of my last relationship; met a woman who I liked, asked her out, got semi-accepted. After a couple months of going out I realized that I was the only one really putting effort into our relationship. Talked to her about it and got a "wait a bit/not serious right not" response. Waited some more and when nothing changed broke it off. I don't regret the time I spent with her, but am much happier now that it's over.
2a) No, you should still try to move it forward, just make sure to give her some extra room and that she shows interest in building or maintaining your relationship as well.
2b) I'm sure they exist, but in my experience are extremely rare.
3) Your guess is probably better than mine.
Your situation reminds me a bit of my last relationship; met a woman who I liked, asked her out, got semi-accepted. After a couple months of going out I realized that I was the only one really putting effort into our relationship. Talked to her about it and got a "wait a bit/not serious right not" response. Waited some more and when nothing changed broke it off. I don't regret the time I spent with her, but am much happier now that it's over.