It's just how it is.

Consequently:

I'm on the first floor. = Az első emeleten vagyok.

He fell from the fifth floor. = Leesett az ötödik emeletről.

I guess you've already been told about this, but I'm repeating it here: Hungarian floors work like British ones. So usually when you enter a building, you're on the groundfloor (földszint), above which there may be a first floor, second floor &c (első emelet, második emelet stb.)

When you have finished moving, you're "on" the floor, not "next to" it (outside the building in the air?).

Not the most common word order, but definitely correct. I can imagine it being followed by "...de tovább nem" -- "...but not any further"