Sorry again for the delay.
The restarts after the crashes are pretty much random; sometimes the computer starts again without a problem and sometimes it takes some minutes for it to function again (although it tends to crash again after 30 mins anyway).
The first time I did the paperclip test the PSU didn't turn on, but then I checked all the pins and everything and tried again, and this time it worked. Oddly enough, while I was doing this my father saw it and asked me about it. I explained him the problem and told me that if we needed to change the PSU he could help me buy a new one, so we ended up buying it (Nox Urano VX 650W) on an internet store.
I installed it and it worked fine the first couple of days, but then the computer started crashing again. The product wasn't refurbished and arrived with the original packaging and all the other stuff too, so I'm not sure if the new one is malfunctioning or not.