Can't get past the opening scene, when the guy says something about something sending us to a watery grave. Pressed just about every button on the keyboard, clicked mouse buttons, etc. The dialog will not advance.

Also, tried running as admin to no avail.

Trying running Setup, to see what the keys are mapped as.
I believe you have to hit 'Action', (default 'A'), to pass.
I believe you have to run as Administrator, otherwise, the game will hang at the end of the cutscenes.
The Setup also has to be run as Administrator to change settings.

I have the exact same issue. My directX is up to date, and I've tried starting the game in windowed and full-screen modes. Windows XP btw.

Yeah, same issue here. Unfortunate (but slightly comforting) so see others are being effected as well. We were discussing this issue in another thread, while it didn't help us, perhaps you could try some of the suggestions from it?

http://www.gog.com/en/forum/treasure_adventure_game/action_key_stops_working