I'll do my best to answer some of these questions.
I think I need to preface my explanations by noting that, in the United States, the authority on rabbit raising and husbandry is the American Rabbit Breeders' Association. The association provides a breed standard for each recognized breed, which details (point by point) what the rabbit's conformation, coat density, length and color, and other characteristics should be.
Why is it so important to know the genetic history of the rabbits you're breeding?
Every breed is recognized in specific colors. There are also colors that are not recognized. Knowing the genetic history of your rabbit gives you better insight into what the rabbit may carry or pass down to its offspring. As far as coat colors go, genetic history is purely a cosmetic characteristic but very important if you plan to show.
More important (and practical) are the genetics that control body type. Conformation controls everything, from the way the rabbit carries its weight to the way the head and teeth develop and more. When you're working with purebred rabbits, you know exactly what you're getting into - it's right on the pedigree.
Some lines are known for certain traits - maybe strong shoulders, but tend to be weaker in hindquarter. Also, not every bloodline crosses with others well. A good breeder will learn which lines mesh well so that the rabbits they're producing have stronger conformation in each generation.
When you're working with rabbits of no known history, there is no way to know what you're breeding. This may be okay if the breeder is willing to cull terminally for unwanted characteristics. But if the goal is to place culls in pet homes, the breeder should have knowledge of the history of the rabbit to pass on to the customer.
What about the health history?
I think this part is pretty self-explanatory. An example I can give is one of our very first rabbits who was purchased from a pet store. She appeared healthy and rambunctious as a young rabbit. But as she reached about 5 years old, her feet started turning into her chest. Her conformation was "off," so the distribution of body weight on her front limbs was too much. The twisting and arthritis progressed throughout the rest of her life. She lived to be 10 years old before she passed away from cancer, which she may or may not have been genetically predisposed to. By the end of her life, one foot stood almost parallel to her chest.
If selecting for healthy, hardy animals isn't a concern, my question would be why is the person breeding their animals?
What about rabbits with poor conformation? Pinched hindquarters for example, and any other reasons.
This ties into what I've already said. Correct conformation isn't purely cosmetic. Breed standards are made to encourage conformation that will support the animal throughout its life. It's very difficult to get all the right traits on one rabbit. As soon as you have a baby born with a nice, full hindquarter, another trait goes backwards...maybe they're light in bone. So you start focusing on heavier bone. Finally, the herd has heavy bone but seems to have narrow shoulders.
It's impossible to produce a perfect rabbit. Raising them is a selective process in an effort to balance as many good traits as possible. When you ignore conformation altogether, these traits disappear very fast. Within one generation, you can start seeing pinched hindquarters again or light bone.
With domestic rabbits, there is no natural selection. There's no "the strongest survive." It's our responsibility as breeders to select for animals that are built for the long lives they live in our care. Desirable traits don't happen by accident, we have to purposefully breed for them. Otherwise, poor traits will take over quickly because there is nothing in a domestic animal's environment that will take them away.
What can happen if you breed rabbits with unknown histories?
How are we bettering the breed if we do this?
I think sometimes the cliche "bettering the breed" is used too loosely. Picking up a pair of rabbits, regardless of their quality, and breeding them isn't bettering the breed...but it's often used as a little quote on websites to justify breeding programs.
Improving a breed is a combined effort among the entire community of registered breeders. As one herd improves, other breeders are driven to catch up. This continued competition is a fun way to meet new people and enjoy a weekend together, but it also moves the whole breed forward. If everyone is concerned about eliminating malocclusion in Holland Lops, the gene pool will become less and less prone to genetic malocclusion, for example.
Is this not how the breeds were first made?
Mixed breeding, you mean? Yes, breeds were mixed to create new breeds. But new breeds didn't come from one person who owned three rabbits of varying types.
Creating new breeds is an extensive process. It takes a large number of rabbits, precise records and a specific plan to get started. You'd probably be looking at a few decades worth of breeding before you have a "prototype" consistent enough to begin the acceptance process. Then the acceptance process takes many years too.
An example here are Lionheads, which are now breeding more consistently and have been around for many years. They're still not an accepted breed, and may still have a long way to go. Even after all this time, they don't breed true on the whole. Most of the consistency you see is within those large herds, operated by long-time, dedicated, and experienced breeders.
I respect the enthusiasm of those who wish to create new breeds. But sometimes I think people see it as easy as breeding a Mini Rex to a Holland Lop and creating a little lop with rex fur. That's not exactly how it works.
Why cant it try and make new breeds by breeding mixed breed rabbits?
You could do that. But like I explained above, it's not common that people with this goal in mind are serious enough about the end result to follow through with it. Creating a new breed (as in, a breed that eventually breeds true and becomes recognized by the ARBA) has the potential to take most of a lifetime of work, really.
Lionheads have been in the U.S. since the early 2000's and were worked in Belgium and Europe for a long time (from the information we have) before that. Possibly decades before what was originally believed. As of now, they have two more years to present at convention for the chance to become a recognized breed. Failing that, the whole recognition process starts again. I think the presentation process can take 3-5 years total, if I'm remembering right.
What about breeding pet store rabbits? Or rescues?
Nope, nope, nope. I will reconsider my "nope" if someone can offer a valid reason to breed either group.
Brother and sisters?
You can breed brothers and sisters, but it's risky. Linebreeding closely can improve traits to some extent, but there is also the potential to double up on poor traits. In my experience, brothers and sisters often have very similar strengths and weaknesses, so they simply won't make a good breeding pair.
Will breeding a brother and sister make a kit with eight legs and four eyeballs? No.
How can we ensure high quality standards and the health of rabbits that are being bred? What issues and health problems/concerns can arise if people don't research these things?
It's impossible to ensure anything 100% when you're breeding animals. As much as we know about genetics, it's also a very mysterious thing. Recessive genes can hide for generations before a problem pops up unexpectedly.
The importance of having a history of your breeding rabbits and keeping records is to lessen the risk of pulling undesirable traits into the gene pool. We can't know for certain what will show up in the nestbox, but we can know for certain that we took every precaution possible to prevent problems.
Whew! That was a marathon. I hope it helps.