I got the game to work by right-clicking the "Configuration tool" and selecting "Run as administrator." I checked both options (Logo and Opening) under "Disable Movies." Without administrator rights, I was unable to save my configuration.
Huh, strange. I actually tried that and it still refused to save settings or run. Only windowed mode worked.
I know on the Steam forums, they have a list of common fixes for the game, but I think having a thread dedicated to all known fixes for the game would be helpful here, as well, since not everyone wants to go to separate sites for game support, along with how slowly GOG gets updates (a problem that might still affect our version might be fixed and a non-issue in the Steam version, for example). On top of that, it seems like most people only really recommend one or two fixes out of how ever many.