1> Dropcloth under the litter box -or- [maybe] litter box with taller walls.
1.5> Cleaning the tile? I got nuthin... I wonder what vinegar or isopropyl alcohol might do. [test on inconspicuos area...]
2> Mix the Muenster with the IAMs; make him work to pick out the food he will eat; probably he eats less/loses weight.
3> I am convinced that Cats are smarter than they let on. See below:
A relative of mine used to have a cat. The cat had trained my relative to do "tricks".
That cat enjoyed sitting on a porch behind a screen door, smugly surveying all that he could see and smelling the great outdoors. Unhappily, there were two doors between the cat and his spot on the porch behind the screen door. So first obstacle, the living room door, a wooden interior door: cat walks up to the closed door, puts his paw on the carpet, rotates his wrist 180, knuckles now brushing the carpet, slides the paw underneath the door, choink! claws extend into the soft unpainted wood on the bottom of the door, cat backs up, door gets dragged open, cat moves to next door. (The 1st door's latch does not work, it is misaligned or something; old house...) Nobody taught the cat that the door latch was loose or any of the steps in the foregoing procedure.
Cat wanted past the door and figured out how to get past it all by himself.
But that left the second door, a metal exterior door. No problem: the cat uttered a simple command, Meowww-r-r-p, Meowww-r-r-p, a couple of times and simply waited for his trained human [my relative] to come open the second door. Voila, porch access...
...for a cat with a plan; and a trainable human.
Does your door have a lock? Can you hang a key on the wall for humans to use?