Here's a hint: notes never work. The best you can hope for is a polite refusal, and then she'll go back to her friends and be like "Hey, this guy asked me out with a NOTE today, how weird is that?" And then they all will laugh. The worst would be if she reads the note and doesn't even realize that you were the one who sent it, given that you supposedly have not even talked to her, and thus she likely does not know you by name. Or that she will be incredibly creeped out by some guy she does not know "loving" her and basically stalking her from a distance.
Also, if you don't know this person at all, A) what makes you think you're actually interested in her? If you don't know her, all you know is her appearance. And that's not enough to build a relationship on. And B) What makes you think she would have any interest in you? As you yourself said, you don't even know if she's single. Most people don't just jump into a relationship with the first stranger that asks them, and those that do you likely wouldn't want to date anyways.
So here's what you do: Grow a pair. You say she sits right beside you in study hall? Great. Next time you're in study hall, talk to her. Introduce yourself. Make it a no-pressure kind of thing, don't worry about asking her out immediately. Just talk to her, get to know her a little. After a few conversations, if you're feeling comfortable, ask her if she's seeing anyone. This is a nice, "safe" way of breaching the subject; It's innocent enough that you're not blatantly implying that you only want to know so you yourself can date her, yet there is the hint that that could be the case. If she gets awkward, odds are she's not interested in you. If she answers quickly and the answer is no, you're safe to go for it.
Alternately, if that's too much of a "risky" move for you (and from the sound of things, it might be), you can ask her what her plans are for the upcoming weekend or what she did over the previous weekend. If the words "boyfriend" or "date" don't come up anywhere in her explanation, she's likely single. And asking about the upcoming weekend gives you an opening for asking her out ("Well, since you're not doing anything on saturday, would you like to come <blah blah blah> with me?").
Another good way to get to know someone is to find out their AIM/other instant messenger name (from them, so as not to be creepy) and talk to them over that. It will help you talk in a more comfortable, less pressure environment, and make it easier to communicate (or at least this can be the case).
Also, as for if it's "worth it" to ask her out with this being your last year and all, there are a number of factors to consider:
1) How long of a relationship are you expecting/hoping for? If you don't think you'll last the 9 months it will take for your first year of college to start, it's not a big worry. And given that this would be your first relationship, I wouldn't bank on it lasting forever.
2) How far away are you going to college (assuming you are going, otherwise this is a non-issue)? If it's fairly close (community college, etc), then time isn't really an issue, as you can still date after graduation.
3) Assuming your first answer was "Very long-term" and your second was "Far enough away that it'd be a problem", how capable do you think you are of maintaining a long-distance relationship? I would not suggest having one with someone you have been dating less than a year (and I have experience in this area).
4) Also assuming your first answer was "very long-term" and your second was "Far enough away that it'd be a problem", do you really want to go into college without any relationship experience under your belt? It'd just make it that much harder to find one in college. Being in a relationship make it a LOT easier to get into subsequent relationships, as you then have that confidence boost from that previous relationship telling you that you are capable of doing so.