Those beeps do mean something, if you count them.
Yes, I tried looking that up on Google. The system uses an Award BIOS, Unfortunately, different sources give different meanings for the same bleeps as the code is, according to one source, defined by the motherboard designer rather than Award
One long bleep seems to mean a memory problem, but I suspect that it is caused by the uncommanded shutdown so I'm not too sure about how much credence to put on that especially as the memory seems to work OK the rest of the time.
Checking the BIOS, it's quite old (5 years), would it be worth updating it?
I still suspect the power supply so will probably replace that.
GG