Sure they can. There are criteria to follow when diagnosing FM and it's so commonplace these days that a competent and studious physician (which of course describes the entire clan) should have no trouble. However, in some cases, when in doubt or if the picture looks slightly complicated a patient could be referred to a rheumatologist, neurologist, or pain mgmt specialist just to be sure. Nothing wrong with that.
General medicine. A general medicine doctor is commonly referred to as an internist or family medicine physician. Each doctor may have different areas of expertise or supplemental training. Most should be able to accurately diagnose fibromyalgia. A good second opinion in a complex case would be given by a rheumatologist.
Yes. ? if "general doctor" can dx fibromyalgia (FM). Yes. If chronic widespread pain, depression, restless sleep, chronic fatigue, confusion & no provable disorder is FM. No tests prove FM, but highly suspicious if potentiating cause(s) (femaleness, obesity, mechanical injuries, hypermobility, inactivity from prolonged illness) reported. Subluxing SIJ, trigger points, TOS & TMJ, & scoliosis are common.