Most of the time if you're having issues like that with the starter motor it's because the starter was improperly installed. The starter gear is bronze, the flywheel teeth which the bronze gear engages are steel. while it is possible to break teeth on the flywheel, most of the time the starter is just not properly shimmed.

Autozone will check your battery for you for free, a dead battery will give you similar symptoms to a dead alternator. It could be something as simple as a loose or corroded ground wire.

It sounds like what you really need is an actual mechanic, as opposed to a parts changer.