1. I'd dive in again but this time I would use the canned air to clean and replace the heat sink compound on CPU and if this is an older PC the GPU and BRIDGE CHIPS.
2. As to the shutdown, research how to run CHKDSK on the next boot up. I missed which OS so I can't offer a search for you.
3. It's proper for a CPU fan to run when the CPU is busy or under load. Some folk forget that.
Yesterday my processor fan starting running non stop after a couple weeks of it running fast occasionally. I took things apart to clean it to check if that was the problem. When I hooked everything back up and powered it on I got a beep code. It consisted of 2 second long beeps that just kept going until I disconnected power. I made sure everything was tight, reseated the RAM, tried again. The PC powered and booted no problem. I got a pop up telling me that my system had recovered from a serious system error. I ran a disc check, that was fine. Now after a few minutes of running the processor fan is right back to running constantly. When I try to shut down it acts like it's going to, I get the "Windows is now shutting down screen" and then it just stays there. I have to physically shut it down every time.