Skarsnik and Gobbla's action has you deal one damage to each zone of an opponent's capital for the cost of removing three damage from the card. Then you draw a card.

If one of my opponent's zones is burning when I trigger the action, is the damage still dealt to that zone, then canceled due to the zone having no hit points remaining? Or is it simply not dealt? If not dealt, does it prevent the card draw, since the "then" clause precedes it?

Whu? No one likey the phalic-hooded gobbo and his pet mouf n' legs? You know you're all dying to know the answer to this question like I am, but are afraid to answer 'cause it's one of them tough ones. Right?

I would say you still draw a card based on the FAQ's Crone Hellebron ruling.