I haven't seen the show, but I do know something about snow caves:
My question is about the snow cave they've dug for sleeping. It was a mistake to keep the large entrance wide open, is not it? As far as I can recall you have to seal it as best as you can leaving just a very small opening at the bottom for ventilation.
You're correct - a small opening located lower than your main cave. Also, make additional ventilation openings - just poke a skiing pole or whatever through the wall.
But if it is warmer outside (i.e. not that much colder than the melting point of water) then the size and shape of the opening really isn't that important. Just block enough to keep unwanted drafts out.
They were also suffering from a headache in the morning, most likely because of the smoke from the fire in front of the entrance entering the cave all night long.
Headache could be dehydration, too... but sleeping in a smoke filled cavity certainly seems like a bad idea.
But there is another thing to notice: Warming the snow cave is totally counter productive - The temperature inside the cave will never be warmer than a few degrees above the freezing point of water
no matter what you do. Excessive heat just means that the walls melt faster.