dude, for only like €300 you could get some heavy upgrades.
So you don't really have to be that rich to upgrade that computer xD
I remember where I couldn't even fork out $50 for computer parts, as "sad" as that might be, I was probably one of a few people who couldn't just pull money out of nowhere :p
Anyway, I quickly read through some posts, but have you tried running a CPU or GPU stress test? That alone would quickly (and aggressively) find any issues with your hardware (in terms of errors, or heat, whichever comes first).
I recommend Prime95 for a CPU stress, and Furmark for a GPU stress. I would also recommend a temperature monitor program too, and for this I recommend HWMonitor from CPUID.
Run Prime95 for like 3 mins, see where your CPU temps go (anything above 60C might be of slight concern, but you may have to look up safe temp values for your particular CPU). If your computer stays on without problem during those 3 mins, close Prime95. If your computer shuts down, this is either from overheating, or your PSU cannot deliver enough power to the CPU (if the PSU can't deliver the power and your computer shuts down though, I would replace the PSU right away. If anything, your computer is supposed to BSoD if your CPU doesn't get enough power).
Next run Furmark. Safe GPU temps are card-specific, but I think a general number to stay below is 80C. Do burn-in tests for about 3-5 mins or however long you feel is safe (but I recommend at least 3 mins). If your computer lasts without problem during the time you run Furmark, good. Otherwise, you'll either get a BSoD, your GPU driver could restart, or your computer might shut down.
I would probably run both programs overnight just to be safe later on though just to check overall stability (but only after you have some "safety measures" in place, such as having your BIOS shutdown the computer if your CPU temps reach a certain value, or have Furmark shutdown after the GPU temps reach a certain value, both values should of course either be very close or at the max recommended temp for your hardware).
TLDR: Stress test your GPU and CPU and see what happens
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