Tried both versions that are installed (normal and "Safe Video") several times, but even after waiting through the unskippable intro cut scenes countless times, when I finally get "control" of George in the plane, I get nothing. There's no mouse cursor, the only thing I can do is pause the game with the Esc key. And to add insult to injury, I can't even save at this point to avoid the intro cutscenes!

Running windows 7. Tried changing to a few different screen resolutions, but no change. I am able to open my inventory with my Xbox controller, but still can't do anything.

Any suggestions?

As far as I know, Broken Sword 3 doesn't support mouse control: it only supports keyboard and pad and they both can be remapped, but are probably mutually exclusive (meaning that they won't probably work simultaneously). Probably the analog sticks of your pad can't be assigned to the movement, though: unless you use a specific utility like Xpadder.

It's not a technical issue, it's the first puzzle, actually.
Release the seatbelt first, then you'll be able to properly control George.