Turns out, it's all Avira's fault. I have been using it and PowerDVD together for quite some time, but it's the only thing on my system that can auto-update without my knowledge.
I reformatted again, and this time, the first thing I installed was PowerDVD. It launched, but would not play DVDs. I then went through a grueling process of trial and error, installing stuff, then testing PowerDVD, then rebooting, then testing PowerDVD, and so on. First came the LAN drivers. Then SP3. Then more Windows Updates. Then more Windows Updates. Then more Windows Updates. Then NVIDIA drivers. After that and a reboot, PowerDVD would play DVDs. After installing Avira, PowerDVD refused to launch. Disabling Avira Guard did not help. Killing the processes did not help. Disabling the startup items did not help. But as soon as I uninstalled Avira, PowerDVD worked again right away.