The official answer used "หรือเปล่า" which felt a bit more like "or not?". I always felt the "ไหม" question indicator was more apt here, but perhaps I'm wrong. Can a Thai person can share their thoughts on what the more natural way is in Thai
Is ที่ the classifier for หมอ or is it a way of making it "are you a doctor who is good"?
Yep, the latter. For a similar example:
คุณเป็นเพื่อนที่ดี = You are a good friend