Supposing I have 1 orc unit in play, and I play Mork's Teef Ritual.

If my opponent responds by somehow killing my only orc unit, what happens?

Do I still get to put an orc into play? Do I still have to deal the damage to my capital?

I would say that the sacrifice is part of the cost, and happens as soon as you play Mork's Teef Ritual, so there's no way for your opponent to kill you unit before it is sacrificed. Therefore, unless the opponent can respond by discarding all units from your hand, you can play a unit of your choosing at the resolution of the tactic (and take the damage to your capital). If you don't have any units in hand at the point of resolution, I guess you won't take any damage.

herzogar is right, Mork's Teef Ritual specifically says sacrifice the unit and the sacrificing cannot be prevented.

Do I still get to put an orc into play?

Actually you get to put A UNIT into play. Any unit.

Personally I have a few bloodthirsters in my orcdeck for that occasion