It is hard to apply classic genetics when you are in a world of magic and magical manipulation. Genetics imply in evolution and evolution imply in the complete logical destruction of all pre-made creatures and whatnot. That aside, you also incur in the problem of turning magic into some sort of deus ex machina for medieval biotechnology - "Hey, lemme use this spell to grow a steel skin in this lineage of Skorgs" - and in the background, dozens of genes are created out of thin air to allow cells to bioaccumulate iron and carbon nanoparticles in orderly fashion on the outer epithelium. Such changes would promptly generate a reproductive barrier. Promptly. Finally, chimps and man are rather close, so were we close to our cousins of the largely extinct Homo genus. That doesn't mean crossbreds between man and chimp or man and Homo neanderthalensis were/are/will be possible. Minor, stupid changes can create reproductive barriers. And tackling your example, even dogs aren't all compatible with each other. Their breeding distribution resembles a horseshoe, with the extremes being unable to reproduce between themselves.
Summing up, while it is quite possible that if you apply genetics you can end up with a justification for crossbreds assuming 'races' do have a genetic compatibility - no matter how improbable that would be seeing the large differences between one another - you are still exposing yourself to the side-effects of introducing so realistic a concept in a unrealistic world. Making it even more simple: it won't be logical.
I agree up to a point that there is little point trying to apply genetics to a fantasy world, however my point was that for those who wish to a pug looks more different from a Grey hound, and a Great Dane for example than a man does from a chimp. But all three dog types can be interbred.
As an Aside if a human & chimp / can or cant be interbred has never been fully answered a Russian scientist did try some experiments but since his work was considered highly unethical and stopped no definative results are available, it's not something that's been tried often (at least not openly for obvious reasons). However othordox thinking is that we are not similar enough geneticly. Would you recognise a human/chimp hybrid if one existed?
Interestingly dogs show an amazingly wide range of Phenotypical variation while maintaining genetic compatability.
You could also argue in a magical world that magic did not alter the genetics at all simply distorted the way that it was expressed and that this distortion is passed on by means other than the genetic code.
Anyway I tend to agree any use of genetics in a universe in which magic exists is speculative, hell men in the world of elemental may not even have DNA for all we know.