Oh, man, this is magnificent. A new non-euclidean room, a new possible juice/game mechanic, a situation that lets us actually practice jumping instead of flying, the symbolism of the door representing a conclusive* departure from Mrs. Slitherss's cruel and inefficient rulership, I could go on.
I have to admit I am biased. Seeing someone else give devil's advocacy to Mrs. Slitherss's relationship with the player is reviving.
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If I may make one suggestion, it would be that the door doesn't necessarily have to directly tie into any great evil or like in the Juice Galaxy world. instead, I like the idea that the door's focused purpose- and frankly this whole dream realm if you'll pardon me- would be the player character's own reflection on defeating, and hurting, the closest thing he ever had to a 'parent' figure in his life, and juggling this with the personal growth he has found since he left her harsh oversight.
On account of this, I like your latter suggestion that this realm would be found by going back into the schoolhouse and sleeping at a desk. It makes more sense for an introspective dream sequence, *and* it gives the classroom a purpose aside from just being our spawn point for a new game. I personally like for significant things and places to have more than one purpose in a game, and it has long irritated me that such iconic places of first impression, such as the inside of the Juice School, often lose their interactable value after being used once, twice, or for a brief time.