H.W.
First, I have two friends who have their doctorates (non-medical) which I lovingly call Dr. (last name) because they worked really hard to get them.
1. Yes.
2. I would follow their lead when it comes to titles. If someone introduces themselves as "Dr. Brown", that's what I'll call them. If they introduce themselves as "Cynthia", I'll use that. Not unprofessional-- esp. nice in the medical business that someone is trying to build relationships. However, when I call for scheduling, etc. I'll always use the full title (Dr. Brown) or when talking to other medical staff about that doctor.
3. Absolutely not-- but if I knew they were a doctor, I would use that title and not revert to Mr/Mrs/Ms. to begin with.