How does an otolaryngologist treat an ear infection (otitis media)?

It depends. Otolaryngologists usually evaluate and treat patients who have had resistant or recurrent ear infections. They treatment they recommend will be highly dependant on each individual person's history, but may include antibiotics or surgical intervention.
Puzzling question. They would likely treat it the same way your Family doc, nurse practitioner, Physician assistant, etc. Most ENT's in my town don't see people off the street for otitis media, they only see patients referred in by primary care docs with chronic or resistant cased in need of specialty care.
Depends. That depends on the type of infection, the symptoms and severity of the problem. It is hard to answer that question without more info.