No, yes. Eye doctors focus on diagnostic and therapeutics for the eyes, but eyes are manifestations of systemic disease so they can treat infections and other things. Based on a physician's comfort level and access or lack thereof to specialists, they will do other things.
Of course! An ophthalmologist is a medical doctor (md or od) and a surgeon. We treat everything from glasses to injury, infection, cataracts, retinal detachments etc. Much more than just infections.