The "Teacher" badge is given for: Answering first question with score of 1 or more. However, the OP accepting an answer is significantly different from a general viewer up-voting a question. I suggest a new badge, say "Troubleshooter", awarded the first time when your answer is chosen as 'accepted answer' by an OP.
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I think this is a very good idea. If nothing else, it should restore the symmetry between the starter badges for asking and the started badges for answering questions.
I would call the badge something other than "Troubleshooter", though. I think the name should be along the "Student"/"Scholar" lines. Since the badge for the first upvoted answer is called "Teacher", I would suggest calling the new badge "Mentor" or "Adviser".
I think it's a good idea. This also teaches about the site features. There is a lot of different types of badges that can be awarded yet there isn't anything on first accepted answer? It's be nice to show a noob accomplishment.
Also, the category would belong to Answers.
I don't think this is a good idea.
I think the general rule is that badges add value to the site if they either
a badge for the first accepted answer wouldn't fall in either of these categories IMO. It wouldn't encourage any new behaviour (You will strive to write good answers anyway), and it wouldn't help you explore new functions (because you're not the one doing the accepting).