Most ear infections (I think the stat is something like 75-85%) clear up on their own without antibiotics, but that if they don't clear up within 3 days or get worse then antibiotics are needed.
A single incident of ear infections doesn't mean your daughter will be more likely to have them in the future. What causes them can vary. She may have had a cold already or allergies and the fluid got into her ears and then became infected. They don't happen with every cold or sinus infection.
Anyway, the amoxicillan should work. If not you'll know within a couple of days due to your child's behavior. If her symptoms don't go away or they get worse, then take her back in. That would indicate that amoxicillan either is the wrong med or the dosage isn't strong enough or she needs to be on it longer or the infection is viral rather than bacterial (and if it's viral then an antibiotic won't work anyway).
If your daughter feels better after finishing the meds, then you shouldn't need to bring her back for a check up unless the doctor requests it. For now just let her rest, give her plenty of fluids, and let her eat when she's hungry.