Let me also try to answer the questions:
So here are my questions:
1.When is it appropriate to email a professor (that you don't know) a question about their work? (Remember; I'm an undergraduate, so these questions have a high chance of being inane in one way or another).
If you have a research level question about a professor's work or related to his or her field of interest it is appropriate to email.
2.How should this email be formatted? Do I give my name in the first line? Or do I give a bit of background, ask the question, and sign my name. (The latter is what I have been doing).
You can first introduce yourself. You may give the background to the question. Try to be clear and rather brief.
3.How specific should the subject line be in order to increase the chances that the email gets read?
As informative as possible, "A question about convex bodies", "Are there only finitely many smooth toric varieties" "An idea related to the Erdos-Faber-Lovasz conjecture".
4.Should I mention anywhere in the email that I am an undergraduate?
Yes. If this is a good research level question it will will be impressing and if not it will not harm.
5.This one is kind of silly, but I always have trouble deciding what to do: Suppose that I've emailed a professor, and they email me back with some answer... is it appropriate to email back with just a "Thank you"? I always feel like I'm wasting their time with such a contentless email, but at the same time I do want to thank them... Actually, in general I feel as though my questions are a waste of the professor's time (probably what feeds the email anxiety)...
Yes. email back with "thank you".
which brings me to:
6.If you happen to be a professor who has received such emails in the past; are they a waste of your time? Please be honest! EDIT: Small new question...
I have received quite a few of emails from undergraduates. Usually it is not a waste of time.
7.If you receive a response and the professor signs with their first name, are you supposed to refer to them by first name the next time you email them?