Barring any healing from Med Teams or other sources, can you trigger War Machine's ability twice? As in, when the damage sustained from triggering it would defeat him? I'm not sure if the ability would finish resolving or not, and I couldn't find a definite answer in the Rules Reference.

As long as you've paid the cost, the ability's effect resolves regardless of what happens to the source card. That's how cards like Indomitable or Tenacity work (you have to discard them to use their abilities).