[beyondloadedSE]

Your car is dying because youve got an obvious short circuit at the radio. Because it stays on when the car is off, its continually draining the battery. Try this. Pull the fuse for the radio under the driver side footwell. If you have the owners manual, all the fuses are listed in there as well as what they pertain too. If the problem persists after you do this, then you have possibly an alternator problem AND/OR the alternator relay is bad.

Btw, the rotten egg smell pertains to a bad catalytic convertor which wont cause the symptons your saying.