FWIW..not much, probably...lol...
I grew up in a family that took in strays, ferals, and rescued any lost furry thing...wild or domestic. We nursed the young, won most and lost a few. The two Yorkies I have now are the first purebred dogs I have ever owned, and both are re-homed and I doubt either of them came from a show breeder that DNA tested for anything. Both were 2 in March 2012, born 6 days apart, neither has a thing wrong with them...Mia is golden, so "not to standard" but ''Geez Louise" is she gorgeous, sweet, lovable, and all Yorkie just without the blue. Brody is a little champ, he strikes a 'show pose' naturally, and is adorable doing it because you just don't expect to see that and...boom...suddenly there it is...and he stands there in it for a bit, and goes back to adventuring.
Oddly, or I guess it is, in all those years of rescuing, I never had serious genetic issues with any of my dogs. I never heard of LP, AAI or collapsing trachea, or any number of things I've heard of since I got these Yorkies and began researching the Yorkshire Terrier breed in depth....for the fun of it and to learn something new that might help me with them...although I fell in love with the breed many years ago and with their personality as exhibited by a little Yorkie-Poo. My Yorkie-Poo of the '60's was a lost, tiny puppy out in the cold, freezing, and was healthy at quite an old age until the car hit him after someone let him out while I was not home.
I would not hesitate to take a puppy or dog, or cat, from the pound. A lot of very good dogs end up there through no fault of their own...and some, like my Shep/Lab Honey did once, because she was a "Houdini" and could get off of and out of anything but a spiked choker collar. We brought her home again, of course...hence the spiker...lol but I just wanted to point out that 'cheap/free' dogs are no more likely to get serious or DNA related health issues than an 'expensive in purchase price' dog...IMHO...probably not less likely to either.
OP, I think you might want to do a few searches on breeds to determine which might be 'right' for your family since you have a choice at the moment. You already seem to have an idea what will fit into your family...just not the specific breed names. Then, if you choose from a breeder, rescue, the Humane Society or the pound, you will have a better idea what you are looking for and what you are looking at.
The AKC website has breed information, and there are many other doggie related websites, too. Your children will enjoy the search as well, I think...and may not be as saddened once they start looking at pictures and reading about the breeds. If you have specific traits and sizes in mind, you may be disappointed if you end up with a breed without those specifications.
I rescued a female mixed breed that had been wanted to be a guard dog but she was too friendly, so the owner decided not to feed her or give her water. When several of us told the owner he would be turned in for cruelty if he continued the practice, he turned her out into the street...half dead. All those other people willing to turn him in were NOT willing to help her, though, and were going to send her to the pound...where, in her condition, she likely would have been put down. She couldn't keep food down when I got her, but when I re-homed her, she was healthy and strong again. She
loved little boys...not little girls. The high-pitch squeals set her on edge. She chose her own family and they fell in love with her instantly, too...but they knew from talking to me a lot about her before they laid eyes or hearts on her. She was no $1,000.00 purebred dog but she was worth a thousand times her weight in gold to them and them to her.
When you get to choose...choose with your brain first...a dog or puppy is
not an impulse purchase...your heart will find its own way, and, for goodness sake, do try to find the dog that chooses you. Rarely will you be disappointed this way, regardless of breed. My opinion only...but from personal experience.
Good luck...and let us know how it turns out. Pix are great!