Are "Doc" and "Dr" both abbreviations for doctor?

Doc: A casual, impersonal term of address
Dr: Abbreviation for doctor
http://dictionary.reference.com

Doc used in front of a name is usually a nickname, or a reference that the person used to be a 'proper' doctor. It's not a formal title. Dr can be any doctor. Note that this includes doctors of legal stuff, literature, philosophy and theology. If you're wanting treatment and can't find a medical one, avoid both legal and theological. Legal you can't successfully sue when it goes wrong. Theological will save your soul (or argue you to death) but not worry about your body. Literature will probably at least have read about what's wrong with you. Philosophy will help you to come to terms with it.

"What's up Doc?"

https://en.wikipedia.org/wiki/Bugs_Bunny