Is there no distinction between 'warm' and 'hot' in Norwegian?
If context doesn't tell you which one it is, you can ephasize it by saying "kjempevarmt" for hot.
is the word "sauna" not common as substitute for "badstua"?
They're both common.
Isn't the 'd' in badstua silent? I can recall an explanation like that in my norskkurs, but i could hear the audio pronounce it quite distinctly.
Yes, I would leave the 'd' out when pronouncing it.
Why is it 'varmt' and not 'varm'?
Because you start the sentence with "det" which is the neuter form. It's the whole gender thing. En rød bil - Et rødt eple.
I said, "Det er varmt i badestua," but it said it was wrong, and that saunaen is correct.
Because it's badstua not badestua, it's without e. The mistake had nothing to do with varmt :)