Motherboard issue symptoms

Oh yeah... I'm way too sober for this one. lol

Sounds like a similar issue that actually plagued the original Xbox consoles. That issue was in relation to contaminated console motherboards. I am not saying it's the case but maybe the board of your xbox 360 console has corroded. Not saying this is the
case but the symptoms of the console turning on and off randomly are similar.

My initial guess when I hear the problems you're facing is that there is an issue with your console. Symptoms sound like a battery on the motherboard is out so when it powers down it's not saving your settings. Do you have it set to Energy Saver or Instant
On? I would recommend to reaching out to Hardware Support and see what options you have at the Microsoft Device Support page. Is this a new issue? In it was saving your settings
previously, but has recently stopped, or has this been a persistent issue with the console since you've got it?

Ok, but can it be a GPU issue or not?

It's entirely possible that it can be a GPU issue, sure. The fact that the system faults when RAM is put in a specific slot points to a motherboard issue (to me anyway)...could even be a faulty motherboard that is also causing issues in the PCI-e lanes which would, in turn, cause GPU issues.

Is it actually freezing it making it so you can't do anything else until you restart it? It might be on its way out, hopefully it isn't. A couple of years ago when the Halo wars demo was out it would always freeze up the Xbox on the same exact spot, never
got to play much of it. Then, bought the game when it came out, and again it kept freezing, eventually got the E74 error, the motherboard or something was broken, no idea why it had those symptoms, though.