Does ophthalmic antibiotic ointment need a dr prescription ?

Yes: Ophthalmologic antibiotics require a prescription but you can get them from your optometrist (od), ophthalmologist (md/do) or family doc (md/do). Best to get evaluated so you can be assured of what you're treating. Not every pink or red eye is due to an infection, much less a bacterial one that can respond to antibiotics.

Yes: It does, like any antibiotic. And make sure it is ophthalmic if it is going in the eye.