Did you try a good canned air cleaning?
Do all fans look to spin at good speed? (that is, they look like they are spinning like new?)
The CPU is only part of the puzzle but for me the GPU is where all frame rate issues start and end unless there is something very wrong elsewhere.
You did fine with your testing but I don't see any mention of cleaning or case cover off testing.
CPU: AMD FX-8120 Eight-Core Processor (3.1 GHz, 16 MB)
CPU Cooler: Coolmaster Seidon 120M
GPU: Gigabyte ATI HD7870 OC 2GB 256b DDR5
RAM: DIMM DDR3 8GB 1600MHz Kingston (Kit 2x4GB)
HD: SEAGATE Barracuda 1TB SATA/600, 7200RPM
MB: Gigabyte 970A-DS3 970 AM3+, DDR3, SATA3
PSU: SEASONIC (Energy Knight) SS-500ET
OS: Microsoft Windows 7 Professional Edition Service Pack 1 (build 7601), 64-bit
BIOS: Award Modular BIOS v6.00PG
So let's get right to the issue. I start up my computer, load windows - everything is fine and smooth. Doesn't matter what I play, it could be Skyrim or Minecraft. The game runs fine for around 10 minutes with proper FPS. Then it suddenly drops to 1 - 20 FPS and stays there for the entirety of my gameplay. I quit the game and whole computer is laggy for certain period of time (until it cools I suppose).
In summer FPS drops much sooner. I played in winter, it was cold outside and my FPS dropped. So what I did is I opend the window to let the cold air into my room and the computer was fine the entire time. So from this information I assumed I have cooling issues but CPU temperatue stays below 50°C as well as GPU. And I even swapped fan CPU cooler for a water one - didn't work.
I don't have means to try and swap out all components one by one to detect the bad one.
Ask me anything and thank you for your answers.