"If you have DNA from strikingly different breeds of, say, dogs, ..."
I'm not a biologist, so I might be mistaken. I'm inclined to say that in theory, yes; in practice, no.
Most biologists would say that the DNA does determine the breed, but that there isn't any easy way of getting from DNA data to conclusions about which breed. There isn't a nice mathematical formula from one to the other, so getting breed information from DNA information depends on consulting a huge database which does not currently exist.
Reading the remainder of your post, I think you do already get this.
Fundamentalism - the anti-American, anti-Christian branch of American Christianity