I've tried the F1 approach which takes me to character selection but past that I can't seem to do anything else. No controls work, F10 doesn't pull up anything. Super frustrating, first time I've had issues with a GOG purchase.

This post: https://www.gog.com/forum/mortal_kombat_123/what_are_the_keyboard_controls
says it's f1 for player 1.

F10 opens setup screen in MK1 and 2, but you may be pressing it too early. Make sure you at least see the Acclaim logo.
In MK3, once you see GT Interactive logo, press Escape to get into he setup menu.

I have the same issue. No controls work other than the F1.

Same here. I can't even select a character. F1 and Alt+Q to quit work, but that's pretty much it. The game doesn't even come with the manual.

Just tried MK3. I can get my guy to move, but I can't get him to kick or punch. Adjusting the keys doesn't work, either. Again: no manual.

So glad I bought these!

Possibly these might help:

https://www.gog.com/forum/mortal_kombat/gamepad_problems_dpad_moving_by_itself_with_mortal_kombat_1/post13

https://support.gog.com/hc/en-us/articles/213040105-Setting-up-a-controller-in-DOSBox-games-even-if-the-game-doesn-t-support-a-joystick