How awesome that you are considering rescue! I have been volunteering at a no-kill shelter for almost six years and have worked with many, many dogs. Most rescue groups and no-kill shelters will have more stringent requirements for adoption (home visits, vet references, etc.) as well as higher adoption fees. County shelters (your regular ol' "dog pound") should have more minimal requirements and lower fees. It would definitely be a good idea to get your cat up-to-date with the vet first, though (sounds like you're planning to do this).
As for a surprise inspection, I have never heard of that. I think they would likely schedule something with you. Another great thing about no-kill shelters and rescue groups is that some will let you take the dog home on a test run to make sure everything clicks. This is in the best interest of all parties involved and helps the organization make a good match. They should also be able to help you test a potential dog with your cat.
Do you know about Petfinder.com? If the Sheltie doesn't work out, check Petfinder for other shelters and rescue groups in your area. They have tons of listings. Something else to mention is that there are many purebred dogs available for adoption. People are often not aware that there are rescue groups that specialize in specific breeds.
After working with shelter dogs for a while now, I have seen some with issues, but the vast majority of dogs I have worked with are even-tempered, loving dogs. It's a myth that all shelter dogs are bad dogs or "broken". Many of the dogs in shelters that do have issues are due to the fact that they just got dumped off at a big, scary place and are confused as to why they are there. They need time and tlc to readjust.
One thing to note, some people think that the adoption fees for animals from rescue groups and no-kill shelters are high. These organizations put hundreds of dollars into the care of many of their individual animals and don't begin to recoup their costs through adoption fees. When adopting an animal from these groups, the animal will mostly likely have already had "the works" as far as vet care goes: up-to-date on shots, any known illnesses (worms, etc.) will have been treated, and already spayed/neutered. County shelters will have lower fees, but the animal will likely not have had this much care.
I will also make an unsolicited recommendation for adopting older dogs. Many people want puppies and young dogs, but they are also the ones that are more apt to get into trouble, pee on the floor, chew things up, etc. Older dogs (and I'm not talking old, maybe 4-5 y.o. or so) are often much more relaxed.
I will end with my testimonial. I have a 10 y.o. mixed-breed shelter dog who is probably a hound/herder mix, and she is just the greatest dog. We adopted her from a county shelter at 1 y.o., and she was quite a handful at that age. After some training and learning on all of our parts, she has since become the love of our lives.
Sorry this was so long (and full of random info)! Could you guess that dogs are one of my favorite topics? :)