Is it a bad idea to ask professors questions weeks before the class starts?

Would it be a bad idea to send an email with questions about the material before the class actually starts?

First of all, I don’t mean to sound harsh, but generally it’s a bad idea to be a person who asks disingenuous questions not because they want to know the answers to those questions but “to show [someone] that they are motivated” and because they “would like to develop a bit of a relationship” with the other person. It’s dishonest, and the thing about this type of dishonesty is, the people the dishonesty is directed at are almost always better at detecting it than the dishonest person thinks. So this approach tends to not produce the desired outcome, and can sometimes backfire in unpleasant ways. And more specifically, most professors have had the experience of having a student try to impress them in various ways that come across as insincere. It leaves a bad taste, which does not help a while later when the student conveniently shows up asking for a letter of recommendation.

Setting this aside, if you have questions you genuinely want to know the answers to, there’s nothing inappropriate about emailing the professor. Professors often enjoy discussing course topics with students and answering their questions. If your questions sound sincere and not like something you contrived just to show that you have something to ask, I’m sure the professor won’t be offended by your email, and there’s a good chance that she’ll answer it. Whether she will like you more as a result or not, I don’t think anyone here can predict, so again, if that’s your true motivation then yes, it’s probably a bad idea to send the email.

For a moment, change your point-of-view to your professor's situation.

She receives an email from a future student asking about some topic that she'll most probably cover in her lectures. This student doesn't give her the chance to present the topics the way she wants, but ignores her efforts in preparing a course, tries to self-study it, and just wants to use her to fill some gaps.

This might not be the best way to establish a positive relationship.

This might depend on how much you ask and how you ask it. Expressing interest in the course with questions can be a good thing. Ask if the professor can "point you to" a source where you can find the answer. Or, ask if they can suggest some way to think about the issue you are having. But my personal guess is that it is less useful to ask for actual answers.

When you express the question, make sure you indicate you are willing to work for the answers and not just seeking shortcuts.

Some might not welcome such questions and some might not give much of an answer (other than "wait for the course"), so don't necessarily expect an answer.

This won't help answer your etiquette question, but note that Wikipedia is generally pretty good about topics in mathematics. I've noticed few errors and they seem to get quickly corrected.