Right from the start the situation is very bad i.e. many random characters, but after 15-20 minutes on it gets better, but still some random characters, this is the reason why I think it might be hardware-related.
This is most likely hardware related, but without checking the hardware, it will be hard to analyze as there could be a couple of reasons for the problems you observed.
If characters flip, then this indicates that something is wrong with the communication with the RAM, i.e. some bits flip while either reading from RAM or while writing data to the RAM.
So that could either be a problem with the RAM, or it could be a problem with the bus drives, or a problem of the CPU, or a problem with the power supply.
If I had to guess, I would probably believe the latter is the most likely reason because chips only age slowly, whereas the power supply is under permanent stress given the amount of current it has to supply. The capacitors in it age more quickly than the rest of the board.
The CPU gets very very hot, and I decided to leave the case opened. Do you think the situation would improve by adding a CPU fan?
It might. So if you like, add the fan, try again. But don't be disappointed if it does not. There are several things you may need to try.