So the main problem started when my GPU (then a radeon 6500 series) began failing, giving me displaced pixels and visual noise and the like. Then all applications, be it games such as Skyrim or just a program like Word seemed to run slower. I updated the GPU Drivers, The bios, did a complete re-install and HD wipe, and even got a second hand Radeon hd 6870 from a friend. The error still occurs occasionally, the computer is still generally slow, and Skyrim will load but not start a new game. This is the limit of my own knowledge, and I would love some help. I can get the DXdiag file if you need it. Thank you!
Assuming u got 3 1gb ram sticks, fault could be one of those failed. Honestly no idea if a failing ram stick would cause displaced pixels and visual noise, but it will cause pc to be slower if it'll run with it. I recently gave my mothers husband's pc new ram 4GB up from 2GB and it made a huge difference in speed (also minimized 'passive' taskbar stuff he had running for no real reason). If possible, I would try to get my hands on some ram and test if i were you.
He tried that already.... OP I would first test ram and HDD with Memtest86 and an app like HD Tune, just to check. Artifacts (bad pixels) usually indicate a hardware fault. If you are able to get a screenshot of the graphics corruption, that will help to determine the cause.... Is there anything helpful in Event Log?
Like pill monster says sounds like hdd or ram slowing you down. My bet is on the ram since you have swapped gpu and you are getting artifacts. How is the cpu temp? Sometimes that could cause degraded windows performance if its over heating but i do not think that is causing the artifacts!