Batteries are sized based on the cranking load and alternators are sized for the total electrical load. There is no direct relationship between one and the other.

When the alternator dies the car will run off the battery until voltage drops to the point were systems won't function anymore, which is what you experienced. Usually if the alternator quits the charge light on the dash will remain on, but a lot of people ignore warning lights. If the battery was charged while the alternator was being worked on then it will probably work okay. If the battery was significantly weakened by the discharge, it will probably show up as a slow cranking rate or an inability to hold a charge, which will result in low cranking speed or no-cranking if the car sits for a few days, so you should watch for these symptoms and replace the battery if it appears to weaken.

A shorted battery can cause excess current generation by the alternator, but that should not harm a healthy alternator. Shorted batteries can happen suddenly. You can start the car and drive somewhere, then come back in an hour and it's completely dead.

Jump starting a car with a shorted battery will sometimes not work because the shorted battery will not allow the alternator to maintain sufficient voltage because of the large current draw. In this case, disconnect the positive cable from the shorted battery, jump start it, and drive it home on the alternator. Leave all electrical accessories off and don't stall the engine!

Duke