I can't make any progress in the game because I am unable to save. The 'Escape' key is supposed to bring up a menu where I can save but pressing it doesn't do anything. I've pressed every single key on my keyboard, in fact, and not a single one of them brings up a menu where I can save the game. This also prevents me from quitting the game in the standard way. I can't Alt+Tab or Windows key my way out the game, either--I actually have to press and hold down the power button on my computer and do a hard restart just to get out of the game.

I'm slowly getting used to the difficult and dated control scheme this game has, but until I resolve these issues I won't be able to make much progress.

AFAIK, this bug happens, when you load the game in the middle of some specific animations (usually - with the "help" of Quick Load via F9).
So just try not to do it again.
Also, you may use Alt+Ctrl+Del and kill Gothic process in Task Manager instead of "hard restart" with Power button.