You started of by saying that what constitutes a dog is that it can only mate with other dogs. This is due to their genetics which also controls the size and hardware to even be able to mate in the first place. How different does a breed of dog have to be before it stops becoming a dog?
What if a dog was "bred" by natural selection to be larger, have thicker fur, have much larger claws, and be able to stand on its hind legs. Would it still be a dog? Could it be a bear? The evidence in evolution says that bears and dogs have a fairly recent common ancestor. At what point did these two species of canine, who were at one time just different "breeds", change enough to the point that one became a "bear" and the other a "dog". What are "bear" and "dog" other then our own classification of things that look slightly different and do not interbreed.
This is a real example. Now lets go back to my hypothetical example of the mole Chiuahuahs. At what point during the gradation of creating the Chiuahuah breed and its potential future outcomes does it stop becoming a dog?
You seem to be thinking of genetic similarity as this discrete measure of a species. The only thing that prevents two animals from being able to produce a viable offspring is a blurry line of how different their genetic makeup is. Notice that this includes the situation where their genetic information might be compatable but, due to genetic differences that affect their physical characteristics, they are still unable to mate.
You also seem to be inclined to use terms like "dog" as if it was some kind of definition that nature rather than men actually grouped a set of animals.