Assume you connected the HDD as slave in another PC!
All the symptoms of a failing/failed PSU so check that out first. I have had 2 go the same way in a matter of months (not the same PC). Indications were - it doesn't switch on when it should and this becomes more troublesome until it doesn't switch on at all. Your CPU "situation" can be checked out later - try to avoid doing more than one thing at a time. If you are in to a bit of DIY, and it looks like your are, disconnect the PSU from all the hardware connections inside the tower and run a multi-meter around the connections. BE CAREFUL.
Your CPU hardware. Do you mean one of the little plastic bits on the CPU housing has broken off - the bit the "pressure" clip attaches too? If so, nothing to be done here as it's mobo integral so the mobo may be trashed by I don't think it's your problem. Your PC may have been running like this for ages so don't disturb it until you are certain the PSU is OK and begin fault finding elsewhere ;-)