I fixed the problem, kind of.
I tried to remove and re-install the required components, but it ended up breaking the system.
So I downloaded a fresh Raspbian image, wrote it to the SD card, and started from scratch. From the Raspbian image, I followed the instructions here and installed the entire system from source, setting everything up exactly as I needed it to be.
The only thing that didn’t cleanly install was the PPSSPP emulator.
I was able to get eduke32 running, though I couldn’t get my 360 controller to work with it, and it wouldn’t grab the mouse, so I couldn’t use the mouse to control the game. (The mouse pointer simply moved around independently of the game.)
Regardless, it’s a big first step, and I’m sure we can figure out the other problems.