Never get any of those mixes or "designer dogs". They are a scam, overpriced, poorly and cruelly bred, with unpredictable personalities, bodies, and often have poor health. Why is it you came up with this mix and wanted it? Do you just like JRTs and Shih Tzus and wanted to combined them? Or did you see them online or somewhere?
Like someone else said, you do need to be careful having small dogs with young kids, as they can traumatize and hurt small dogs, which can make the dogs retaliate and bite. However, with proper supervision I do think Shih Tzus are a good small dog with kids. Most puppies will be ok with cats if they're raised with them but you do need to be careful with certain hunting or ratting breeds like JRTs that have high prey drives (ie, instinct to chase small fuzzy things).
I think it's also a good idea with your situation to get a medium sized dog from a shelter that's either an adolescent (6-18 months) or adult. That way you know the dog's personality, and it's a bit larger so can do better with a small child. You can find purebreds if you want one, and on the Shih Tzu matter I work in a shelter and there's never a time we *don't* have at least one purebred Shih Tzu! Although unless you live in the NYC area I don't know how prevalent it is elsewhere.
If you insist on a purebred puppy you should go to a good breeder that will show you the mother (and father if he lives there) and kennel facilities, give you extensive health records, and make sure your family is appropriate for that breed/puppy. NEVER EVER buy a puppy from a puppy store or order them from a website or "Breeder" of any sort online. They are puppy mills and backyard breeders and force dogs to live in horrible conditions, and breed unhealthy pups.
Other breeds I might recommend in a small to medium size dog that's good with kids might be a Pug, Cocker Spaniel, Beagle (questionable prey drive though), Bichon Frise, Cavalier King Charles Spaniel (but they often have serious health issues) and Poodles--NOT anything "doodle" just an old fashioned Poodle. Hope this helped!