When you tried to install Windows on the HDD and test stability with that, did you actually go ahead and disconnect the SDD? You'll need to make sure to do that because even with it being present it'll be enough to trigger instability, regardless if the OS is installed on it or not. Also, have you tried with the SSD only? Who knows, it could be the HDD that's bad here.
As for Prime95, how did you run it, and for how long? Prime95 usually should be run for 7+ hours straight on Blend settings, and then an additional 7+ hour run on Large FFT settings. Depending on the stability of either test we can get a better idea if it's the CPU at fault itself, or something in the CPU-to-RAM crosstalk.
Also, when you say that the system starts freezing up once you start using it but remains stable when idle, what activity are you actually trying to do that triggers the freeze? Sometimes it's best to log what you were doing at the time of a system freeze as there may be patterns involved.
For additional hardware testing, I recommend running
Hwinfo on
Sensors only option and set it up to log onto a file and let it continue logging till it crashes. Do this twice and send both logs. We might be able to see if there's temperature or voltage instabilities taking place. Also, if you motherboard has onboard graphics, try removing the video card and test stability by running graphics from the onboard video.
It is evident we're dealing with hardware failure, it's just a matter of ascertaining what piece. Unfortunately sometimes this ends up resulting in a process of elimination and slapping potentially bad parts into a stable system to see if it bugs things up.