It won't run reliably unless it's on the lowest cpu setting imaginable

This M50 will give random BSODs (random error messages or almost random) and it may reboot automatically unless I manage to boot and then set it to the lowest cpu setting possible (lowest cpu speed+optimal/max cooling).

Any help or hint will be greatly appreciated. Since there's no warranty anymore I'll either have to fix it or get a new one, I don't have the time to wait for prolonged remote tech repair support anyway.

It's not clearly a heating issue (unless it started by "burning" something in the past) since I cooled it for several hours being off and still got the same symptoms in mere minutes. I hoovered the fan, same anyway.

It puzzles me since if the cpu setting is low it's very reliable, no errors of any kind (while it's not related directly/solely to temperature). Hard disk runs fine and fast too.

The battery is old and weak but that was the case for 2 years. The ac adaptor is replaced with a third party one but that was the case for 1.5 years and it's CE approved (though I'm thinking of checking its voltage though I'm not confident it'll help any).

The laptop gets easily unstable if it's physically moved, e.g. if I lift it up now and move it around it'll most probably hang (frozen desktop screen, nothing happening, having to shut it down).

I've checked the ram, nothing, even switched the internal chip with an additional I've put, same.

Any ideas? Remember:
- it's not at least in the short term related to heat since even after shutting it down for hours symptoms reoccur in mere seconds/2 or 3 minutes while the fan works fine and it's cleaned(unless something happened to it in the past because of heat)
- it's curiously very reliable if the cpu setting is kept very low. days have passed being stable on several programs with it (though it's tediously slow on that setting) (could it be something on the chipset?)