No, dogs are not part of nature.  They're domesticated animals.  Doesn't take a biologist to see that--they've been subjected to generations of selective breeding (this involves extensive culling of 'nonconforming' pups, BTW).  All it takes is a set of eyes.

Dogs are definitely NOT 'non-GMO', unless you're prepared to argue that Chihuahuas are conspecific with timber wolves.  Mammalogists don't think so.  That's why they classify the former (and similar strains) as Canis familiaris and the latter as C. lupis.

What's REALLY interesting is that the drastic physical and physiological modifications to C. familiaris could initially have been accomplished in as little as half a century.  Consider the Russians' work with gray foxes--Urocyon cinereoargenteus:

http://www.auburn.edu/academic/classes/biol/5090/boyd/Captivity_effects.pdf