Password in user settings is no longer valid
It seems with heat issues, you'll want to make a very good backup at night after it's been cool or off all day and then wipe it and test if you can rely on this hardware.

The software can't possibly work when the hardware is throttling or stopping the processing. Without charging circuitry - you can't really use a portable Mac. Also, it will be hard to separate a shut down due to SMC failing to sense proper charge / voltage and this might have nothing to do with normal sleep.

Looking at this from the outside, I'd start with a clean install to see if you can confirm all the symptoms are directly or related to hardware.