Show Quality chickens are from breeders (good ones) not hatcheries because, well, for starters - Hatcheries originally got their birds from breeders. From there, hatcheries only look for a chicken that lays enough eggs to sell. Also, they often keep their old stock which can get inbred and crossbred. And of course - Sometimes they purposefully let this happen. For example, Araucanas and Ameraucanas sold by hatcheries are actually crossbred chickens, not the breed claimed.
Breeders actually have a look at and selectively breed their chickens for "show quality" which basically is the closest possible to the APA Standard of Perfection for the breed. The standards are available in a book, sold by the APA. In that book is what each breed
should look like, and what each breed is judged by in shows.
I don't know of any case where you need papers or registration for showing your bird - Even feedstore chicks have been entered in shows. They just don't do as well, because they were bred for laying eggs.
As for the diseases and all, I believe you do need proof from someone that they are healthy birds. Since I have yet to go through this process, I do not know the details of that.
Yes, show quality birds are more expensive than a "pet quality" chicken, but they're still the same cost to feed and take care of.
The reason hatcheries do not have show quality birds is, as stated above, because they do not keep track of their breeding. Most hatchery stock are huge amounts of chickens who breed whatever they can, wether it be direct relative or sometimes even a bird of a different breed. And, over time, this causes a lot of faults in the bird. Some in the end look nothing like what the APA standards call for.
Have I answered everything? Trying to see here and I think I have, but I'm not sure.
Oh - Breeders, where they get their stock from. . . Well, from previous breeders, really, if they're reputable. (some are just people who breed hatchery chickens and claim they're good quality though) But it all traces back to the origin of the breed.