On her Incapacitated side, Nuit causes the Environment trash to be reshuffled into the Environment deck. If there is no Environment trash, do I still have to shuffle the Environment deck? The same question applies to Incapacitated Atum and the Villain deck, but obviously it should have the same answer.
There's no real reason to unless you've stacked the deck using Visionary, Nightmist, Wraith, or Tempest. A random deck will still be random whether you shuffle it again or not.
If you have stacked one or more cards (top or bottom) then yes.
I agree. There's no point to the physical act of shuffling if all you're going to end up with is a deck that's just as random as it is currently, but if shuffling it will increase its randomness, it needs to be shuffled.