Hi Human1 and
As you have discovered, these type problems are difficult to troubleshoot because there are so many things that can cause a system to crash, freeze, or suddenly reboot.
When dealing with hardware issues, I always want to ensure I am provided good, clean, solid power. So if me, I would swap in a known good power supply, or have that on professionally tested. A plug in tester or multimeter are not conclusive - especially with intermittent problems.
Being 5 years old, it could be just about anything that is now showing its age and is beginning to fail even under normal loads. I am assuming the interior is clean of heat trapping dust, and you have lots of front to back air flow through the case. You say the temps are good, what temps and what are they? Since you are getting AMD errors, I am assuming that is for your graphics. Are you monitoring your GPU temps too? If you have another graphics card to swap in, you might try that (uninstalling the old drivers first).
Memtest is good, but no software based RAM tester is conclusive either. So running with one stick at a time is a good idea.
Inspect the motherboard for bulging or leaking electrolytic capacitors. These failed or failing capacitors are a common cause of sudden, but seemingly random system lock ups and reboots. The capacitors look like tall soda cans, many of which surround the CPU socket.
All older motherboards, and many of today's less expensive motherboards use electrolytic capacitors containing a liquid electrolyte. Failing (including flawed and/or abused/over-heated) capacitors literally bulge at the seams due to excessive internal pressures. Extreme (and very rare) cases result in a firecracker type explosion or “pop” or “snap” type sound that can really stink up a room. Typically, electrolyte just oozes from the pressure relief point, which appears as a symbol or letter stamped in the top of the capacitor casing. The electrolyte can be caustic to motherboards and flesh. Look for white to dark-brown, dried liquid or foam on the tops or bottoms of the capacitors. Bulging capacitors are a sign leakage is about to occur.
A motherboard with bulging or leaky capacitors can be repaired, but often it is more cost effective in the long run to replace the motherboard.
Be sure to first power down, unplug the computer, and keep yourself discharged by touching the bare metal of the case before reaching in.