Edit: I'm deleting my original guess, and going with a new one.
I'm going to guess Lindblum.
Four parts to my whole,
In the beginning of the game, the available areas are: Theater District, the Industrial District, the Business District, and the Grand Castle.
one gets destroyed,
another cut in half,
the other two remain intact.
The Industrial District gets destroyed completely when Brahne used Atmos. I'm not quite certain which of the other three locations get 'cut in half', but I'm thinking it is the Business District. The Grand Castle and the Theater District remain whole.
One piece has top, middle, and bottom.
The Grand Castle contains the Base Level - the bottom, the Middle Floor, and the royal (Cid's family) quarters - the top.
You may exit the bottom of this piece,
you may exit the top,
or the middle of another which has no other.
Foot, air and foot, in that order only.There are three methods of entering this place: the Dragon's Gate which is a land exit, Serpent's Gate which is a sea port, and the Falcon's Gate which is the aerial port.
The Middle Floor of the castle lets you roam the city, which only has the Air Cab travel - no other method. While the royal quarters lets you use the telescope.
Where am I?
Lindblum.