What kind of doctor is it that deals with a sports/joint/tendon/muscle injury?

Choice of physicians. Typically your primary care physican can treat a soft tissue injury. They however may refer you to a physcian who specializes in these areas. These physcians can include an orthopaedic surgeon, physiatrist or a sports medicine physician.
Many. Many different types of physician deal with sports injury on a daily basis. Orthopaedic surgeons (especially those specializing in sports medicine) and family medicine sports medicine specialists concentrate on evaluation and treatment of sports injuries.