Is there a building material that can be cut and glued, can be used outside, and provides good insulation?
I've built some dog houses using SIPs (structural insulated panels). They usually have about 6 inches of foam on the bottom and 2 inches sides and top. All of the materials are salvaged (except the electric floor warmer). I like to keep the bottom off the ground so that snow melt can't get things wet.

SIPs are a sandwich of plywood-foam-plywood (or OSB). Making custom SIPs for a dog house takes some basic carpentry skills, and a of couple week-ends. 

One of the most common mistakes, in my opinion, is making the shelter to large for the animal. Another is inadequate insulation and/or water proofing, especially the roof and floor.

Good small dog or cat shelters can be made from old insulated Coleman type picnic coolers. Line the interior with carpet and cut an entrance hole in one end. Even the very best pet shelters need to be monitored for temp and moisture. I am glad to share detailed thoughts with anyone interested in the topic of pet shelters.