As a general rule I stay away from shotty parts store alternators & starters. Instead I have a local shop rebuild the old ones with quality parts. In most cases I save money, never had any issues with the work my guy does.
I did not see it before, I apologize if I missed it....but...be careful around that battery if it is hot and stinking. There are large amounts of hydrogen gas right there. IT CAN EXPLODE. I have seen batteries come right off the charger without venting and cooling do that. Let it cool, blow air around it to dispel the gas. Wear safety goggles when hooking the cables up. BE SAFE.
............................Maybe it's just me over-reacting but, I think I would consider replacing that battery now that you may have the alternator/regulator thing worked out.