In general, do shoulder injuries heal well with time?

Depends: Simple conditions such as tendonitis or buristis usually heal with time. A torn rotator cuff or labrum may require surgery if conservative treatment options (ie medication, therapy, injections) don't help. See your physician if pain lasts greater than 2-3 weeks.

Depends: Really depends on nature of injury. A torn rotator cuff or labrum will not heal. They can progress and become larger and more problematic. If the tendon is just inflamed it can respond to therapy and injections and other conservative modalities.