Can a family medicine doctor get certified in sports medicine?

Yes. There are many good training programs in primary care sports medicine in which a family medicine doctor, emergency medicine doctor, or a pediatrician interested in athletic injuries and conditions can train. Despite the publicity of professional athletes getting surgery, the majority of sports medicine issues do not require an operation and can be managed by nonsurgical specialists.
Yes. By completing a sports medicine fellowship and then taking the caq sports medicine exam offered through the abfm.
Yes. Yes, there is a board exam and certification of added qualification.
Yes. Primary care sports medicine is a 1 or 2 year fellowship after family practice residency. It is also available to internal medicine, pediatrics and emergency docs. We then take what's called a certificate of added qualification exam to be boarded in sports med. I completed mine in 2003 and love the specialty. Lots of fun!