A lot of chi breeders have started litter box training their dogs. I think it's an amazing idea for all toy breeds, especially while the owner is out at work
Some form of registration to verify that the doh is pure (if it is being sold to you as such). Stable temperament, working ability included, health, conformation. On the breeder's part, willingness to take the dog back at any given time in the dog's life if need be, be interested in their dogs...
In the offspring? I'll agree with Spot here, that's weird. You can guarantee that the parents are clear, but not beyond that. Just a much higher chance you'll get a healthy dog. But that is not a guarantee
What can be tested for, can be guaranteed. For tests that require testing each time a litter is planned can be guaranteed for every litter separately. You cannot guarantee anything outside of that