I don't know anything about it, but here's what I think:
1. Library? Are you crazy? They'll trace it to your library account (at least they would in my library, there are no guest accounts).
2. If you're using a laptop, isn't there like, an internal IP and an external IP? So they'd find one of them... (I'm probably wrong on this, I don't know anything about comps.)
3. Public place - wouldn't people around you get suspicious as to why you're in a teacher's account?
4. You use a proxy - your school can and will contact the creators of the proxy and ask who was using it at what time, easily finding you.

So don't do it.
Really.