So, is both "La scala è vicina all'entrata" and also "La scala è vicino l'entrata" possible?
That would signify that when vicino is an adjective, it is conjugated according to gender and needs a preposition, but when only vicino is used it is because it is an adverb.
No, when followed by a preposition, vicino is an adverb and does not vary. Vicino a means near to, close to, next to; vicino da means from a short distance; from close up.
Duo's version is modifying the verb. It seems weird to me when that verb is essere, because "is [in a position]" should really use stare, and sta vicino sounds right to me. Maybe it's an idiom or an archaic form? English did the same: "next the entrance" used to be normal.
are you asking if it should be "vicinA" because "la scala" is feminine? i wonder about that too
The sentence "the chair is close to the window" was La sedia è vicina alla finestra" - I think. So where's the logic?
No. La tromba delle scale, from tromba = trumpet. It isn't the space under the stairs, it's the whole vertical space containing them.
I am English. The 'stairwell' is the space under the stairs. The word you need is 'staircase'. But we usually just say the 'stairs'.