Bizarre Wi-Fi connectivity issues
Not much of an "answer", but I eventually tried connecting the machine directly to the modem via an ethernet cable, and turned the wireless off altogether.

This solved my connectivity issues immediately. Since I tried troubleshooting everything I could find online, I am left assuming it must have been a failing wireless card. Unfortunately the Hardware Test diagnostics do not run on my computer (this machine has lots of problems...) so I can't tell for sure without simply getting a new card installed.

If anyone else is ever experiencing the same symptoms I was, maybe this is also your problem.