Theres a map shaped like his head in the community hub. You should pay the entire price of the game and all the dlc just because of this.

I don't think theres an actual doge, but the words goat and doge both have the letter o in the same place, so this is pretty much doge simulator.

There's a head called Herbert that looks kind of like doge, and you can probably make a level shaped like his face or something.