I've just begun my search for representation and have run into a few questions. Many internet sources indicate most agents won't respond to email queries if uninterested. If that's the case, I would prefer to continue my search rather than wait for responses that will never come. With that in mind:
1) Is it appropriate to send a follow-up email?
It depends on if the agent says they respond to email queries. If they do and you haven't heard back then it's ok to follow up.
2) If so, how long should I wait before sending the follow-up email? Would 3 weeks be too soon?
I suggest four to eight weeks, depending on how long that agent says they take to respond to queries (check their website for more information).
3) In the follow-up, is it appropriate to mention that this will be my final unsolicited email?
No. Just follow up.