I love how you're taking your time, asking questions and doing research.
A pet is not something that should be bought on impulse. People wait 9 months for a baby, a pet that will be in your life for the next 12-15 or more years shouldn't be something you just decide and do one day.
I've worked in a very busy vet clinic for over 15 years, I have helped run a pet rescue, I have worked very closely with our local humane society as well as other rescue groups. I will tell you right now I know of a rescue that has 9 pug mixed puppies in both fawn and black that need homes. They are not impossible to find. They look almost all pug but we can't be 100% sure. They are 10 weeks old. They were in the room to be euthanized when the rescue group just couldn't take losing 9 puppies simply because they had ringworm and they pulled them out of the shelter. That's 9 pug puppies that would have been dead if they'd been brought in just a day earlier. None of these puppies have homes and it will take a lot of work to find ones for them. Just as an FYI the rescue that has these puppies will adopt to a home with small children. My cousin just adopted two small mixed breed puppies from them and they have 4 girls all under the age of 6.
A few observations from my first hand experience in the dog breeding/rescue/veterinary world.
Purebred does not mean more health problems. It also doesn't mean less health problems. If you decide that a purebred dog is what is best for your family do your research and then do some more. Do not buy a dog from a breeder where you cannot see where the puppies were born and raised and at least personally see and meet at least one of the parents. Do not take a new puppy home before 8 weeks of age. Yes most puppies are weaned by 6 weeks but those next 2 weeks a puppy learns important dog things. Do not buy a puppy from a breeder that has not taken the pup to the vet, had it dewormed and vaccinated at least once. Learn every single known genetic health issue associated with your chosen breed. Learn what ones should be screened and tested for in the parents and at least 3 generations before that. If the breeder cannot produce this information, run do not walk away from them. The main ones with pugs are luxating patella (knee issues), elongated soft pallets and eye issues. These are genetic and should be avoided in breeding. Also learn the characteristics of the breed. Pugs are not low activity dogs. They are high energy but have several health considerations in regards to their exercise. They cannot tolerate extreme temperatures in either direction, long runs or walks. They can be good family pets but they can be a bit stubborn in their training and often times can be difficult to housebreak. A good breeder will not make money. They will have firsthand knowledge of their dog's linage, will have spent the money for genetic testing of the parents and will always agree to take the dog back if it doesn't work out. Don't base too much on a health guarantee. Often the rules of such a promise are not followed through and the only real option you have is to return the "defective" product you bought from them in exchange for a new improved product.
I have held too many furry babies as life left them to feel right about not trying to explain to you why you should adopt a dog vs. buying one. For your situation you are just looking for a family member. Do don't seem to want to show your dog, improve the breed, nor do you seem like you're stuck to the idea that a pure bred dog is the only kind of that that will make you happy. You sound like a reasonable, wonderful home for a dog that so desperately needs you. I'll try to keep this brief to hold your interest, but there's so many reasons it's going to be difficult.
It's just not true that you can't find a pure bred dog small dog in a shelter. I myself adopted a french bulldog over 2 years ago after only 30 days of looking. I found her 4 hours from my home sitting in a city ran shelter and paid an astounding $60 for a "rare" small breed, pure bred dog that was spayed, fully vaccinated, vet checked and temperament tested.
It's not true that all rescues and shelters won't adopt small dogs to a home with a small child. I am asked all of the time if the dog is good with kids, and I usually respond with is your kid good with dogs? It's a two way street. Even some rescue groups that say no adoptions to homes with a small child, will not turn down a home that has taken the time to prove their child knows how to behave around a pet. A good rule of thumb is any dog and child left unsupervised is a disaster waiting to happen. If you let the rescue know you are aware of this and have taken steps by teaching your child how to be calm and respectful of this new pet sometimes they will bend this "rule." Sometimes they won't, but there's always another rescue or shelter out there that will. I fully admit some rescues have gone too far in screening homes. What happens is they tend to chase people right into the breeders hands. I understand why they may have come to take this stance, but I will never agree with it. The shelter we adopted our frenchie from, had our dog listed as not adoptable to homes with small children. DD was 4 and after talking with them at length they had no issues about sending this dog home with us and our young child.
When you adopt from a shelter or rescue you are not only saving one life but two. Not only is the pet you take home given another chance, but you have also freed up that space to give another pet a chance at a new life. 6-8 million pets are euthanized in shelters every year. A large percentage of these are pure bred dogs, and yet another good number are small dogs. If you do your research, ask questions and keep looking at sites such as petfinder and calling your local shelters you will find the dog for you. If you are willing to travel a small bit of distance your chances improve greatly.
I'll let you in on a little secret about some of the rescue groups. Just because one person in that rescue says no, doesn't mean another one won't have a different answer. Each person involved has their own opinions and since it's volunteer there aren't a real set of rules. If you feel a certain dog is right for your family, call back and try to reach another member of that rescue. This trick doesn't always work, but it does sometimes. If it doesn't, then that just means your dog is still out there.
To be honest if you are just looking for a family member to love for many years I cannot think of a single reason why you should buy from any breeder. On the other hand I can think of 6-8 million reasons why you should adopt.
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The most important thing when adopting a rescued dog don't give it human feelings and emotions. This is often what causes the issues with behavior in rescued pets. The owner feels sorry for them and everything the dog does the human decides it's due to severe abuse and neglect. It might be but if you never tell the dog no, or teach it that certain behaviors aren't allowed it will never learn. I once had a maltese lock onto my pinky finger while helping with an exam. The owner's reaction? The grab and pull on the dog, while it's still attached to my finger, yelling at me to be careful because Muffy was abused in utero. 
. He loves the water and I mean he will choke trying to get in the water even if it is a little puddle, he loves kids even though we don't have any, once MIL was petsitting he got out and went to the elementary school because he saw the kids were out playing outside. He loves to fetch and play with the bigger dog and the cats. He is really a great dog all together. Now as you can see most of the traits my pom has are not from the breed like loving water but he was brought up with a lab so he just went along with what the other dog did. Puppies are easy to mold to the kind of dog you want them to be. Of course they do have their own personalities that son't change but that can be seen since they are very little. 
. We took him in after cleaning him up and a vet visit we learn he has a tick spread disease. He is 4 now and it has not bother him, it doesn't mean it never will but it could. Last year he develop allergies not sure if it is a long term thing since we had a big fire close to our house and the smoke and ashes seem to bother our other dog too. So we will see this spring if it is a long term problem. Of course being part lab allergies are common. Again he is a wonderful dog never had issues. 
