Well, I am sorry to hear that you are having these problems and our software couldn't find anything. After taking all the testing you have performed and the issue you are seeing, the first thing that comes to mind is a possible power issue.
I know from personal experience that if your system is getting inconsistent power (from the Power Supply or from the wall doesn't really matter), it can wreak all kinds of havoc and be very difficult to identify as well. Not long ago my personal system would have all sorts of BSOD's (Blue Screen of Death), and all my initial troubleshooting pointed to either a driver or a memory issue. After repeatedly attempting to diagnose the hardware without success, and still having problems after trying different OSes (Windows and Linux) with different drivers. I finally determined it was the power supply that had failed, and after replacement it has been running like a champ.
Unfortunately there aren't any easy ways to diagnose a defective Power Supply(PSU), and if the problem only appears under load then a PSU tester you can buy over the counter will likely declare that everything is working fine. If possible I would recommend running the system with a different PSU that you know to work, and see if the issue continues.
I look forward to hearing your findings, and hope you are able to find a solution.
Thanks,
Kris