Please don't ask me for a source (because I'm lazy and don't care enough about the topic) but I read an article that explained that dogs have a certain DNA replication quirk that makes them both highly variable and easily manipulable.
From what I remember, their genes were more likely to make multiple redundant copies of themselves within the genome. Even though it was the same gene, differing numbers of it would cause different traits to be expressed. The bull terrier was pointed out as a good example of this as a gene dealing with skull growth was duplicated generation over generation causing their distinct curved noses.
I was curious and checked it, and you're right on the first part, and you got one thing wrong for the last part, else it would also be right (not the gene was duplicated, but parts of the gene within, elongating it basically).
So, you would e.g. in a very simple model assume that one trait is regulated by one gene.
e.g. with blood group, you have gene version #1, you have blood group A, you have gene version #2, you have blood group B or version #3, then you have blood group H/0/whatever-it-is-called (quick check on wiki tells me that this IS the case).
In humans this is for most traits NOT the case. If you wanted to manipulate human height or eye colour, you'd need to manipulate dozens of genes.
Not for dogs. Dogs have a very simple genomic structure, and many traits are controlled by simple mutations on single genes, or by lengths of these different genes. To manipulate the height of a dog, you'd need to manipulate a single gene.
Therefore, it is easy to breed them. If you have 2 small dogs, you'll only get small pups from them, because the trait is on a single gene, and both of them have that single gene. In humans, 2 small humans might have a small kid, but the trait is controlled by X different genes, which interferre with Y other genes, and the combo in the kid also could lead to higher growth (because we just have no idea how this works, because it's too complicated).
In the paper they also found what Narz says: Dogs might have the highest variability in the different traits for terrestrial animals (and obviously they found that dogs are highly inbred).
Source,
https://www.genome.gov/27540744/201...ts-attributed-to-simple-genetic-architecture/ , the source scientific paper is at the bottom (interesting and good, from what I quickly saw).
The case of the bulldog skull gene is a separate paper, see
http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1002849 .
EDIT: MIght need to point out that I'm a biologist ^^.