Battery becoming very hot

Mattax said

Turn on the headlights with the key off. See which way the ammeter moves, if at all. Also see if the cigar lighter volt gage registers anything.

Then turn key to accessory and turn on the heater fan.
Note the ammeter and voltmeter

Click to expand...

Hope you haven't given up on trouble shooting this.
If the ammeter always shows charging with the engine running and never shows flow with engine off there's a good chance someone wired it into the alternator output line. That's my best guess right now.