most likely the alternator has a diode shorted out. every other alt i've seen had that problem when the light was on/key off.

i am concerned about the no-start and the hot coil. granted , ignition sw on, power to the coil, the coil will get hot-normal, if the key is left on for 5-6 minuted or so. if the coil is hot with the key off, then a wiring short or a bad ignition switch might be to blame.

i would, unplug the alternator, key off-is the light off? yes, you need an alt.
no, the light is still on, you have another problem.

unplug the alt and try to start the car while using a booster battery or a VERY GOOD GARAGE GRADE CHARGER. then you can see why the car won't start, no spark, no fuel, etc. good luck, chuck.