Is that a good idea to send an errata list to a professor whose class I'm currently enrolled in?
Unless the professor is a total jerk, I don't see any way this could hurt.  My experience has been that academics in general are quite happy to hear from people who are interested enough in their work to offer corrections.  (This has been the case even with authors who, for whatever reason, don't post errata.)

I wouldn't worry about being seen as kissing up.  But if you are concerned you could always wait to send it until after the end of the term.

Before sending it, you may want to casually mention: "I've been reading your book which I really like.  I did notice a few typos though.  Is there an errata list posted somewhere?  If not, I could make a list of the ones I found and send them to you."

If there is any chance that a given typo is not really an error, but something you have misunderstood, you can be more delicate by phrasing it as a question.  Instead of "On page 34 you forgot to require that X is compact" you could say "Are you sure the argument on page 34 works without assuming that X is compact?  Isn't the punctured plane a counterexample?"