How can I fasten pink polystyrene wall insulation into a structure?
Firstly, as has been said, that material will not hold up to either the cats or the weather. You need an actual siding over it. 

The construction adhesive you mentioned will eat the foam unless it's the water-based "panel" or "project" variety. The solvent in the heavy duty type dissolves polystyrene. 

I built a similar "hot box" for my dog using the same stuff. I assembled six panels by creating a frame of 2x2, fitting 1-1/2" foam inside, and screwing 1/4" BC plywood to both faces. I strategically extended the plywood 2" on one face as needed to create the lap necessary to screw the other panels in place. It's extremely solid and comfortable. I have it mounted off the ground by just the front panel, and I can stand on it with no movement whatsoever.