I have no idea how to ask this girl out

Hi, moonlight543.

First things first: is she accepting of LGBT people? It sounds like she is, since you're not worried about how she'll react to your coming out -- which is a good start. But just in case, think about it.

How to come out: If you trust her reaction to your sexuality will be positive, it doesn't really matter how you come out to her. However, since you're close friends, I would personally do it face-to-face, and when you two are alone.
You could make it into an announcement-type thing: tell her there's something you want to talk about, and then do. Or you could bring it up when a vaguely-related topic comes up in the conversation: if you two are watching a movie in which you find the actress attractive, say that. If you see a girl you once had a crush on, say that. If some politician says something about LGBT issues, talk about it -- you'll find it's not very hard to lead from there to your coming out.

When to ask her out: There's nothing wrong with that dance being your first date, but you can also ask her out beforehand -- that's really your call, and I have no advice to give except pick the option you prefer.
When you ask her to the dance is not going to affect whether she says yes or no -- unless she already has a date. So it's probably better to be on the safe side and not have anyone else ask her first. ^^ Aim for too early instead of too late. Don't worry about looking "clingy." You're her friend, not some quasi-stranger who's idealized her.

How to ask her out: Ah, this one is trickier. Once you've built up the courage to do it (and I'm congratulating you in advance for that), again, do it when you're alone.
You could do this the same way as coming out: either at a pre-determined time and place, when you're alone and neither of you has to stay after the conversation is over (say, not while sitting next to each other in class for two hours). Pick a time when neither of you is stressed or rushed, and a place that either can easily leave. A park, for example. Or just in front of the school building before saying good-bye after class. Say you've been thinking about the dance, and you'd like to go with her. Would she like to? -- Let her answer. If she seems nervous or flustered (like it's a surprise to her and she's not sure what to say), tell her she can give you her answer tomorrow if she'd rather. Don't put her on the spot.
Or you could wait for the dance to come up in conversation. If it's a school dance, I assume some people occasionally talk about it. That's your opening -- "Do you have a date yet? Is there someone you'd like to go with?" And there you have it. You're talking about it, so... lead in to saying you'd like to go with her.
There are probably a million other ways you can go about it. ^^ But that's what I'd do.

On maturity: Relationships are something you get better at with practice, like most things. It's okay if your first relationship isn't perfect; few are. Many many people's first relationships are far from "mature," and that's fine. It's about exploring, discovering, and learning -- sometimes almost as much as it is about feelings.
That doesn't make your first relationship not a "proper" one. You care about her and you want a relationship with her. What's more real than that? Just take care not to hurt her and not to let her hurt you (as much as possible, at least), make sure you two respect each other, and then... have fun. There'll be some mistakes and some stumbling awkwardness along the way, but it's part of the experience.

Dealing with rejection: Rejection hurts. It sounds like you think you could get past it and would want to remain friends with her, which is great. Sometimes, it's not possible right off the bat -- because you're a little embarrassed, or because you're hurt and disappointed, or because she's not sure how to be friendly without giving you the wrong idea. So you're right, it can get awkward. The only thing you can do is to stress that you accept that you're just friends and that you want to keep that friendship. Behave the same way you did before, with one possible exception: don't flirt. Even if it was part of your friendship before, tone it down a bit, at least for a few weeks.
After that, the rest is in her hands. There's sadly no way to ensure it won't be awkward at all -- you can only do your best.

Hope that helps, at least a bit. Good luck, Moonlight!

 

I'm not sure that is the best strategy if you want to try your luck with her. You coming out to her should be an act of trust, and the fact that you have second thoughts can turn things complicated. Good you seem aware of it.

What does she think about lesbians, or at least bisexuals? Is she one? Have you heard her speaking good things about it? Perhaps it could be better if you tried to know what she thinks before. If you are really good, close friends, I'm sure she'll have no problems telling you openly, and then you can plan your next step from there.

Well, she is your friend, so you have a perfect excuse to spend with her as much time as you both want. I'd say go out, have some fun together...

Then you can go together to the party, and on your way make sure you do reserve your first dance. However she is free to dance with whomever she wants, and you should be prepared to see your girl dancing with someone else and not dying of jealousy. Dancing is not the same as marriage.

You don't. You can only study the topic by reading books, listening to people's life stories and think for yourself. The good, exciting part of life is that we make things as we go.

We come back to the first question. It depends on her too, what are her views of LGBT folks. If she is disgusted by them, she won't want you around, so it's better to know what are her thoughts before you risk to end up a friendship.
Also if you come out to her on the wrong moment, or if she doesn't feel ok with you being attracted to her, she is going to run away. I've kept my mouth shout with many girls (let's say 80% of my girlfriends), because I knew they wouldn't want me afterwards, and who likes rejection anyways. I've been always the shy boy at the corner of the party watching the beautiful girl dance with mr. perfect, but when mr. perfect was gone, they came back to me to tell me how wrong mr. perfect was. And I was still the shy boy, but I was the shy boy who could listen to her, and being a friend to her, and to me that was enough... 'Cause when you like someone for being someone, you don't care that someone doesn't love you back. True love never dies, even if is rejected or if it never has the chance to manifest.

I wish you the best... Keep us posted if you want.