I had an art history professor that insisted on being called doctor; she said she'd put in a lot of time and spent a lot of money to get that degree, and so she wanted to get her money's worth.
I was being serious. She made art history--which is normally a fairly dry subject, particularly when you're covering art before 1100CE--a really fun and engaging subject.
No; adjunct faculty can also rightly be called professor without having achieved a doctorate. I've had a few professors that had BAs and MFAs (esp. since I'm not sure that there are PhD programs for fine arts).