Well it sounds more like the computer, perhaps some power standby savings gone amok, or some other mainboard problem. Can tou get to setup/BIOS (used to be F10 when booting) to see what the settings are - especially for ACPI? While it is working, (when it is working) maybe download and run the spyware/adware/virus scanners mentioned for free all over this board to make sure it is not some kind of software malware.
Lastly, do you have a system restore CD which you could use to reformat and reload the entire operating system to make sure it's not that once the settings have been reset? You said you had done so twice, but if the hardware has the improper settings, if that could be fixed first.
This part of your post still is peculiar:
"When connected to another CPU and Monitor - they both boot properly."
I still vote for computer rather than monitor, though.