I Like this girl who I think is extremely shy, what should I do?

Does she have any other close friends that you know of? See if you can get 1 of them to ask her how she feels. If not, then try giving her little hints that you like her. She obviously feels comfortable around you which is great for girls like her,. Trust, me I do the same exact thing. I have this 1 best friend but I never really thought of liking him until his friend told me that he likes me. It helped that I knew his friend kinda well so if you are going to have some1 tell her, its better to have them be some1 she knows pretty well. Don't make her give an answer right away if she doesn't automatically. She might need some time to realize that she likes you. You said she feels comfortable around you, so I'm sure she wouldn't mind going out with you. As for what pace to take it at- go slow but just a tiny bit fast than her, she won't want to make the first move if she's as shy as you say. Another idea is say something about liking her when you're poking fun at each other watch her reaction and you can probably tell how she feels and that way you can pass it off as a joke if the timing isn't right. From what you said, I think she likes you though.

I think you should stick with your relationship with her. Although yes, its probably irritating that she wants to leave just as soon as you arrive she's obviously trying her best, give her a break. She's obviously not comfortable in those kind of situations but she came out anyway didn't she? When your around other people just try to make her comfortable, stick by her side as much as you can. Introduce her to the other people you know, include her in your conversations.
As for being hard to read, if she's as shy as you say, its probably just that she's unsure what her next move is. With a girl like her, I would suggest just being honest as you can be. Don't play games with her, try to make sure she knows that you like her. I know that means putting yourself out there but I think that's the sort of thing she needs to hear to put her at ease in your relationship.

When she is around others, she is more self conscious of what others might think of what she says, and that's why she is quieter. Nothing abnormal there. If she is not like that when you're alone, then I wouldn't be worrying about that so much, but you better understand that she is not comfortable when others are around; I have run into a lot of girls like that.
I would try to spend more time alone with her, and just take your time, you don't want to rush either.