GPU dying? Other possibilities?

Maybe you'll have to wait until next winter before you know what the problem is - but lets find out if there are any other useful tests that can bring more immediate results.

Is it possible to use your computer with the case lid off? If you are able to take the lid off, then you can point the room fans at the computer and blow in cooler air that way.

Download and run HWmonitor | FileHippo. This software can monitor things like cpu/gpu/hdd temperatures, as well as fan speeds.

Use Device Manager to Double check that you have the most recent Device Drivers installed on the computer, for the graphics card.

Can you try a different Video playback software, other than MPC-BE, temporarily? Windows Media Player/Centre will do.

Can you (temporarily) disable Hardware Acceleration in Firefox?

Make these changes, re-run your tests, and upload any results to your next post.

Are you getting any artefacts appear on the screen when the GPU gets hot?

Run dxdiag from the start search box and check each tab for reported problems.

You can upload screen shots to the Forum, take a picture of HWMonitor, and upload. Here is posting instructions Screenshots and Files - Upload and Post in Seven Forums

Zoloft said:

made sure GPU is seated correctly, from a cold start (internal temp of home is 21 C), five minutes into 720p video playback, temp was 111 again, at which point I stopped playback.

Everything here points to the gpu heatsink not doing its job. I would remove it, put on new thermal paste, then re-attach it. It can appear to be seated correctly to the eye, but the underlying thermal paste may still have migrated away. Also, you mention two internal fans but did not mention the gpu fan itself.

you also might consider a new card. That 9500GT is 11-year old technology, so even a cheap used card will be will be much faster

Zoloft said:

Replacing it is definitely an option, I just always have the concern of what would actually fit.

find your motherboard at the link below and click add, and then search for video cards. It will only show you the cards that are compatible with your motherboard. You can sort them by price if you want something cheap. Although I bet the prices for cards no longer in new production were the original price, not today's price.
System Builder - PCPartPicker

then you can go here to see how its performance compares to your 9500gt. If I were in your shoes I'd find something like a used fanless radeon hd5450 on ebay. Video cards die of heat and fanless cards do not produce much heat in the first place (except after 11 years, lol)

Compare Graphics cards

When I bought my Vista system 11 years ago, it had two GT 8800 cards in SLI. One failed after a few years. When I checked at NVIDIA, I found that they had disabled automatic fan control in the driver. Dumber than nails. Then the other one failed and I had to buy a new graphics card. Don't know if that applies to the 9500.