Re: Alternator warm after a month gone.
The two options are pretty much field current and backfeed through the (failed) diode bridge.
Typical full field current at 12V is 5A, so you have a 60W heater if the field current is on. Most likely source for field current when not in operation is a failed ignition switch, field current generally passes through the ignition system only after the engine starts, but a failure in the switch leads to field current all the time. If the battery voltage is below 13V then the regulator has no reason to limit the field current and will let it run forever (unless you have a fancy regulator with some special functions).
The other option is backfeed from the positive connection through one or more failed diodes in the rectifier bridge. Since diodes have an unknown resistance after failure this could be a small current or a large one.
If you don't have a current meter, do you have a battery switch that disconnects alternator positive from batteries? Assuming everything shut down, you can open the switch and see if that kills the heat, in which case likely bad diodes. On the other side, disconnect the field wire (which may be hot) and test the same way. Or test by voltage, field wire should not have voltage with ignition off, if it does something is not right.