My next task was to run Memtest86 and Check Disk. Both utilities reported no issues. That was the first bit of good news I had had from the computer in a while. At that point, the issue was narrowed down to either the video card or the motherboard. I ran down to a local 'chop shop' (that's what I call a store that sells computers and used computer parts). For $20, I was able to get both fans and a cheap video card (RADEON 4600 512MB).
I replaced the rear exhaust fan, but had some trouble getting the case off to access the front fan. I left that for later and kept the case open with a regular desk fan blowing in it. I replaced the video card, fired her up and installed the drivers. Word came up without an issue. Okay. I tried GiMP. No problems. No crashes, no blue screens.
That was already an improvement over what I had been getting. I took a deep breath and started testing my games. I started small with Hearthstone which should be less taxing on the video card than anything else. Hearthstone came up and I played a match (Tavern Brawl - I won!)
There was only one test left I wanted to perform. I needed to see if I could run WoW without a blue screen. In a few moments, I was looking at Honors' face in the load screen. I went ahead and loaded in and checked on my Garrison. The RADEON 4600 was really struggling. I was getting around 15 fps. I went into the video settings for WoW and turned it all the way down. That got frame rates up into the 40s until I went over to Tanaan. I'm not sure what they did in Tanaan but my frame rate dropped back to around 18 to 20fps. It's playable, but it's not something I'd want to do long term.
My main goal was to test if the video card was the issue or something on the motherboard. While I can't rule out that there might still be some issue lurking in the motherboard, everything I'm seeing points to the video card.
My current theory is that something caused the front and rear fans to stop working. It wasn't power because even with a new power supply they still wouldn't spin so it had to be mechanical. Either at the same time or because of fans stopped spinning, the power supply fan stopped spinning and the power supply overheated. Either the heat or bad voltage from the power supply damaged the DDR memory on my video card. Whenever the video card tried to access that part of RAM, it would blue screen.
The new GTX 1070s are running about $500, but I found a deal on Amazon for a new desktop AMD FX-6300 for about the same money. It's only got an R7 260, but that's still better than the RADEON 6870 I was running. That will get my computer up to the recommended specs for Legion and I won't have to worry there's some lurking issue on the motherboard I haven't found yet.
If I need to, in a year or so, I can save up some money again, and those fancy GTX 1070s I'm drooling over will (hopefully) come down to a more manageable price point and I can upgrade.