I don't know much about Chinchilla behavior, but I can tell you that chinchillas get hot, not cold, easily. Just look at their profuse fur. It's warm enough to keep them cool in the Andes mountains. It can definitely keep them warm in the generally mild climate of America (perhaps too warm).

I'm not a fan of keeping chinchillas as pets, personally, because it's gotten to the point that chinchillas are nearly extinct in the wild but profuse in the pet trade. That doesn't make sense to me. Plus, they're naturally shy and, when let loose, they hide and run in all directions. The chinchilla we keep at our zoo is not let out in any room with hard to reach crevices because chances are that they will end up stuck in there. I've heard of chins getting stuck behind washing machines, in little pipes, etc.