Addendum to further testing:

Going off of the advice of Square-Enix tech support, the following potential fixes have proven unsuccessful:

-Updating all Graphics Drivers
-Updating all Controller Drivers
-Unplugging all non-essential hardware (everything except Keyboard, Mouse, Monitor, and Power Supply)
-Disabling all antivirus and firewall software.
-Setting all Final Fantasy XIV programs to Windows 7 Compatibility Mode
-Disabling all software that uses internet access.
-Running Windows Update, then restarting.

I have also been told by tech support that FFXIV is unsupported by Windows 8 and 8.1, and that it could cause problems like this. I have also been told that the game is incompatible with Xbox 360 controllers as well, which, again, may cause problems like this.

I'm not going to give up, however, but I'll keep testing whatever I can to ensure that this very serious issue is dealt with.