Do all mouth abscess require being drained?

All abscesses : Require drainage for proper healing. Regardless of location.

Yes... See A Dentist: Yes, treatment is required. Antibiotics may be necessary to resolve the pus which you are describing as a white pimple like growth. An abscess usually becomes more painful as it seeks to find a way out. At some point it will break open & spill the pus out into your system. You need to resolve this asap & see a dentist as the bacterial load will not be healthy for you & is potentially serious.

Yes: Most oral abscess will need to be drained and treated, and not just with antbiotics. You need to remove the cause of the infection inorder for it to heal completely.

No, not at all: Seeing a dentist is best. The dentist can evaluate the location and the size of the abscess. This then determines if antibiotics alone is needed, if antibiotics and some kind of cleaning procedure is needed, and if surgical intervention is needed.