Chances are if the proc was toast (In most) cases it would not get even that far. Granted this isn’t 100% but it’s damned hard to kill most modern procs.
Some basic questions:
When you installed the fan did you use thermal paste, or did it have tape?
Did you clean the surface of the CPU prior?
What temp is it running at in the BIOS after it’s on for awhile?
If thermal paste was used how much did you use?
At this point I’d suggest running diagnostics once it’s verified that is okay.
This a free memory diagnostic, granted it’s memory it can be used to determine CPU, and motherboard issues as well based on the manner errors are provided.
If no errors occur it could be a possibility some data may have been damaged or improperly written or that there is another hardware problem present (Based on the fact you can’t properly install. In which case some different and more complete diagnostics might be able to pinpoint the issue.