At times the system will turn itself on in the middle of the night.

and

Occasionally it will try to connect to the internet,

Definately, definately flash that BIOS, or at least pull the battery on the MB and short the posts. IIRC, That will dump the memory, and it will start up off jumper settings or off default settings and re-load the flash memory BIOS from hardware ROM. Or at least thats what you could do about 4 years ago. If that trick works with whatever is out there today, then you'll have to update the BIOS. MSI and others have protocols for doing that over the web. Potentially risky, but wat the heck isn't.

If you plugged that HD back in to that system w/o doing something to the Flash Memory BIOS, then start over is my best advice if you want to be sure. Its not a vector common in viruses or worms, but if you got attacked by someone with skill who owned you that deeply without even you knowing it, then anything feasible is possible. Plus, unless the system was merely sleeping, power-up from hibernate or deeper imples something system level, not OS level, at least in my humble opinion.

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