1) The most important issue in dog breeding is breeding dogs of good behaviour.
2) The second most important issue in dog breeding is health.
In particular I feel that two questions are important in breeding and keeping dogs.
Question 1: why are we breeding dogs? what for? work? companionship? a mix of both?
Question 2: what do we need to achieve the goals set in question 1?
There are dogs that are bred and kept 'for work'. These dogs are expected to have the right temperament and physical characteristics for the work they are bred for. People breeding exclusively work dogs seem to be a pragmatic lot unencumbered by pedigrees and breed standards – all I ask from them is never to give a working dog to people looking for a pet, because the needs and temperament of these dogs makes them unlikely to suitable as pets, especially for people with little experience.
All other dogs are pets. They might do some something useful from time to time, or even often, but are primarily, or solely, companion animals, because even if they never do anything useful, or are not that good at the ‘job’ they are meant to do, they will still be kept and loved by their owners or breeders. Because the keyword is 'companion', I believe that the most important part of breeding these animals is behaviour. A dog's behaviour does not simply affect the owners – barking being an example for all other stuff dogs do that have an impact on third parties. Alternatively, one might not mind the fact one’s dogs like to dig holes in the park, other people might not be so keen. Dog training is a big business because either people choose the wrong dogs or because even reasonably easy dogs are too much of a handful for some people. Breeding dogs who are as easy to manage as possible should be the first consideration in their breeding. Obviously nobody can do miracles, because some behaviours are hardwired in dogs, and can only be managed, not eliminated. Dogs will always bark and dig and chase, but dogs whose behaviour can be more easily managed are better pets than dogs that require more effort. In addition, if pet dogs are by and large biddable and well behaved, if they do misbehave we can all blame the owners, not the dogs
For the second issue, health, I like to look at two different facets of this topic. The first are extreme and/or clearly unhealthy phenotypes (short faces, bulgy eyes, angled hips, etc). Dog breeding is plagued with many just so stories, where breeders just believe that a specific phenotype is the best for whatever reason, with little or no connection to reality or the health of their dogs. To give an example, a Rhodesian Ridgeback breeder in a famous documentary about pedigree dogs mentioned that these dogs have a ridgeback (and thus a high risk of dermoid sinus) because those dogs with a ridgeback ‘were the best lion hunting dogs'. This is figment of the imagination: if lion hunting qualities are important, it is necessary that the dogs are lion tested at every generation, to choose the best lion hunters. In reality modern Ridgebacks are not lion tested (and to be honest, we are all better off for this), so we cannot tell whether the lion test would, or would not select for the ridgeback. So, these dogs are pets and as pets it would be advisable to breed them without phenotypes with a strong association with disease. I could go on with these post hoc stories (lax ligaments in Newfoundland dogs are necessary for swimming; short faces in bulldogs were selected so they could keep breathing while biting a bull; and so on and so forth) – far too many for me to really have the time to provide details – that breeders come up with to justify some extreme and unhealthy conformation issues. There is no reason whatsoever to breed dogs whose conformation is the source of health problems and in fact it is perfectly possible to breed dogs with a great temperament that have a good sound conformation that is NOT unhealthy.
The second thing in dog health is genetic diversity. Dog breed have existed for thousands of years, and for good reasons: if we want a dog for work or as a pet it is easier if we have a good idea of how a puppy will grow up rather than just hoping for the best. Nevertheless dog breed used to be bred as ‘landraces’, not as pedigree breeds. That is, if a dog had the right temper and physical characteristics for whatever it was meant to do, the dog was part of the breeding pool, irrespective of origin or parentage. Because in the olden days most people and thus most dogs did not travel very far there were different regional variants, though the breeding pool was never closed. In modern days we have a large number of dog breeds kept as a closed pedigree and following a precise phenotypic standard. The level of population fragmentation these cause is astonishing: for instance some breeds differ from one another just for coat colour or nation of origin. These artificial limitations are pointless, negative and based on no scientific reason whatsoever. Thanks to some judicious crossbreeding and backcrossing (if needed at all!) we could now breed dogs in a way that is more in line with the old landraces while retaining breed differentiation (just in case people have not noticed it’s not that pedigree dogs bred to a standard are not still under some form of directional artificial selection – otherwise all pedigree dogs within the standard would have the same chance to be used for breeding, something that is patently false).
Finally, I encourage everybody to get a solid grip on themselves. Plenty of people live happy and meaningful lives despite having or developing a number of health problems at some point in their lives. If a dog is a pet, we need to balance the benefits that its companionship will bring versus the inevitable hassle, and possibly disease. It is perfectly possible to breed dogs as breeds (or landraces) that are sound in body and behaviour, and healthy, or at least with no more health complains as we would hope for ourselves. The overwhelming majority of issues in dog breeding are perfectly avoidable using some common sense and some well understood scientific principles. I just fail to see why we should not achieve this goal.
 because dogs can have an effect of third parties, and because some people have an irrational fear of dogs, any dog owner in their right might should try and avoid creating situations which can be used an excuse to ban dog ownership outright or micromanage it through legislation, neither of which seems to be a desirable outcome.