A HOT alternator means one of TWO things:

1) Bearing or other failure, causing friction - friction means heat.
2) Far too much current is flowing - this would be due to low resistance between the alternator output and ground - this is commonly called a "short circuit" - but I don't use that term.

Since the light is on , it implies the alternator is not charging, but this may or may not be the case. Your voltage readings imply an issue. At 2500 RPM on the engine, the alternator should be able to sustain 14.2 or more volts. But if there was current flowing, it would drain the battery quickly once you let the car sit. That amount of current would take the battery down in way less than, say, the usual 16 hours the car may sit if it is a daily driver.

The battery light circuit is rather simple - given all the indications, and not being there to verify for myself, I'd say the alternator is bad. If you change the and the light does not go out, then I am wrong.

At NAPA, you can pick up a rebuilt unit for $220 or so. Sounds like a lot of money to prove me right or wrong, but I don't know what else you can do.