Wow, I've not seen anything like this. It sounds like some corruption in either the registry, which is not unlikely, or on the system board itself.
It's rather difficult to nail down stuff like this, but here is what I would do if I had the system in my shop.
I prefer to diagnose the hardware first, before chasing down software issues. I can't count how many times I've seen a bad ram chip cause unbeleivable issues that people spent days chasing with software fixes. Most computer stores want to reload windows as the first thing off the shelf, and although this often fixes things, it's my last resort because often times you lose something that is a pain to recover.
So with all that said:
First step: backup any data that you don't want to lose.
Next determine the brand of hard drive in the system, then go to the manufacturer's website and download the free diagnostic software for the hdd. If you have a problem finding this, just let me know and I'll help you out. Some hdd's don't have their own diags.
Next download and create a bootable cd of MS Memory Diagnostics
Run the hdd diags first, it doesn't take as long, and let it run the extended or long test.
Run the memory diags next, might want to let it run overnight, and when it starts up, hit the 'T' key to run the extended tests, let this run for a minimum of 2 complete passes, and preferably 3.
After that, then my next suggestion would be run sfc/scannow to make sure there are no damaged windows files.
Then open a command prompt and type: netsh int ip reset c:\resetlog.txt
This will reset the tcp/ip stack.
After that, remove them again and reboot to see what happens.