Is it harmful to try to learn an advanced topic that is way beyond your mathematical maturity, even if you're really interested in it? Should I focus only on standard material for undergraduates and leave the rest to grad-school?
It would only be harmful if you found it frustrating and dry.
Mathematical maturity is important. But there are a few kinds of it. You gain maturity by becoming familiar with subjects. By learning problem solving techniques. By coming to grips with "unmotivated abstraction" -- learning to figure out what a mathematical construct does without somebody giving you the "magic words". You'll need all of these, at the right level, to get a good grade in a class.
But you can gain -- especially in dealing with unmotivated abstraction -- if you study beyond your means. Learning to ask the right questions is a very good thing! Also, a lot of mathematics comes down to memorization -- memorization of definitions, theorems, and techniques. You might not be able to memorize many techniques if you can't try them out, but you can get a big head start on the other two.
But, like I said, stop if it gets boring or too frustrating.
I know I'm late to this party, and I more or less agree with what nomen said. However, it happens that this is related to pretty much the only advice I've ever found useful about studying math, so I thought I'd share.
Ravi Vakil [who happens to be an algebraic geometer] has an advice page which answers your second question strongly in the negative. However, he doesn't recommend "studying" way-over-your-head material in the same way that you are used to "studying" for a class.
Having learned a lot of math this way myself, I'm fully in favor of this advice. I also think it can stave off the boredom or frustration that nomen talks about, so it is somewhat practical to learn this way.