I think I got most of my terms from working at the Inn (boarding facility).

A- I would normally call that a cage, but I sometimes call them wire crates. We didn't allow these at the Inn.

B- I call that a crate or carrier. If a dog was used to sleeping in their crate we would take the door off and allow them to keep it in their room at the Inn.

C- I would call that a run. If I were going to call anything a kennel it would be this.