So I hoped everything would work out, and that the problem was just the alarm's temperature being set too low. Unfortunately this isn't the case. It got to 167F at one point and froze. Here's a better look at the temperatures:
Here it is sitting idle:
Here it is with the siren going off:
The fans do slow down when the computer is idle, but they're going full blast when the siren goes off. Is a 30 degree jump normal? It seems a bit excessive.
A 30*C jump is not normal. Go into BIOS and turn off all Fan-speed options - keep them running full-blast all the time until temps reach a safe range. Where do you keep your case - open with airflow around it or in a cubby area with limited flow? I'd open your case, look for anywhere to put a front intake fan to get some cool air entering, and make sure your rear fan is free of blockage.
While the case is open (if you're confident enough to do it - sometimes it's nerve-wracking for a 1st-timer) you may want to pull apart the system. Meaning, unhook all the wires, pull out the RAM, graphic cards, PSU, MoBo, etc. and clean them from every angle using compressed air. I'd also take off the HSF and CPU (again, I know you already did), remove all traces of thermal paste, then reapply (spend 5 minutes with it, no such thing as too-OCD with paste, I say). Just for the sake of this Thread and your situation, apply a very thin layer of TP to the entire CPU heat shield, then put another small dab in the middle of the CPU - this will ensure the best degree of heat-transfer. Reinstall the CPU/HSF while the MoBo is out of the case to get better angles on the locks.
If you feel comfortable, remove the back panel on the PSU and blow out the insides with compressed air, too. Not a huge biggie, but if you've gone this far, it's only an extra few minutes.
Regardless of if you pull everything out or not, it would be very wise to do some cable-management inside the case. Basically, spend some time and bundle, group, run, and zip-tie the PSU wires, HDD, Front Panel wires, etc. in a way that they affect airflow the least. Here's a pic of mine:
If after all that you still see such high temps, I'd opt for a better HSF than the stock one you have now.
Sorry this is taking so long to fix. Usually things go quickly here, but sometimes troubleshooting like this is the only way to go. Does anyone else have input on this or see anything I may have missed?