Is emergency medicine the only docs in medicine that work night shifts/weekends?

No. There are many doctors besides emergency medicine specialists that work night shifts and weekends. For example, many family medicine practices are open and have coverage on weekends. There is also usually a specialist on call for certain circumstances. For instance, if a patient comes to the er with a broken bone, an orthopedic specialist will often come in and take care of that person.
No. Almost every specialty works nights and weekends and others come in emergencies. Some (but not all) of the specialties that work many nights/weekends in addition to em are hospital medicine, internal medicine, family practice, pediatrics, ob/gyn, general surgery, trauma surgery, neurosurgery, cardiology, pulmonary/critical care, gastroenterology, and neurology.