Test and Measure.
With no specific errors other than system instability, sometime testing and measuring is the most effective way to drive out a hardware issue.
First you need to be able to reproduce the problem on demand. If you cannot reproduce the problem, then you will have a difficult time with the measuring part. Systems that crash every 1 hour to 6 days are the worst!
Change the hardware in ONE way, then try to reproduce the problem.
First remove ALL unecessary peripherals. Start with Motherboard, CPU, Memory (One stick), Power Supply, Keyboard, Video, and Monitor. Test with each memory stick separately. Swap video cards. Add a drive and load with only one drive. Swap any parts possible. Try a different OS install like windows? Check for cooling problems. Make sure all drive connections are good. Swap cables.
The hardest are CPU and Motherboard since most people do not have these on hand and they can give extremely odd and elusive errors. I've had times when I could not even reproduce the errors on demand very well which made things near impossible to measure.
Point is a motherboard and CPU can be the problem and your symptoms are indicative of something to do with CPU/motherboard. If it is not hardware, maybe a BIOS setting? Definitely low level.
I've had write errors on hard drives that were power supply related; lock up errors that were CPU related; Memory errors and drive errors that were cooling (motherboard chipset) related. All of these solved through making a change and testing. Sometimes that required buying a part to test and measure; had to buy the CPU and Power Supply on faith.
Just a note, be careful with tests; like I said, I had a memtest throw errors on the memory. Then I had seatools throw errors on the drive. I assumed that not ALL of those components were failing at once and found that the heatsink on the chipset was sensitive to dust build up and air flow. So just because you have memory errors, it is not necessarily bad memory. The same goes for other tests.
In the end when all logs and errors run out of indication, test and measure. Good luck.