Hi Patty - let me first apologize for the novel you are about to read.
A bit about me: I'm a total dog person - professional trainer for a few years now and I've been involved in showing/breeding/rescue for about 18 years. Yes, Steph, you are right! I do breed dogs on very rare occasion.
There is really no way to predict what a mix will be like. They could completely favor any of the breeds in there, have all the best traits, all the worst traits or anything in between. Labs SHED like mad, Shelties are big barkers typically and Shih Tzus can be pretty tough nuts. All are relatively easy to train with the Shih Tzu being the most independent in there.
FYI there is no such breed as a "wire haired Shih Tzu". Shih Tzus are a long coated breed - they grow hair and don't shed. Wire coated dogs typically have very hard coats of short to medium length and typically shed quite a bit.
The things you are concerned about "barking, snivelling, shedding, table shark, that won't heel" are, with the exception of shedding, pretty much all based on training. Any dog will heel if trained to do so and any dog will eventually counter surf given the opportunity.
Una, I will respectfully disagree with you on mixes vs purebreds. A -well- bred purebred is very likely to live a long healthy life and have a great temperament. A badly bred purebred is a HUGE risk. Mixes are just a toss of the dice - anything goes. After all, mixed breed dogs are mixes of purebreds! Their genes didn't mutate in transition. They are all dogs!
The main benefit (well bred) purebred dogs offer is predictability of type. That is, if I am getting a -puppy- and I know I want a dog of a certain size, coat type or general temperament or to do a specific job I am more likely to get that than I would from a random mix breed. This does not mean that they are "better" than mixes. Just easier to predict - that is the whole point of purebred dogs.
Patty, if you are set on a puppy and your son's allergies are a big concern, I would recommend you to get a purebred. What a great sadness it would be if the dog is hard on your son's allergies and you had to rehome him. Adopting an adult dog would be a fabulous option - You'll know what you're getting right away! If you are willing to do what it takes to handle things if the dog does cause an allergic reaction, go for the puppy.
I wold also -very strongly- urge you, if you are going to pay money for a puppy to do your research. CERF, OFA, thyroid testing, etc should all get performed on ALL breeding stock as well as conformation or performance testing even if you yourself are not interested in that at all. It shows the breeder cares about producing healthy dogs that are able to perform their breed function. Good breeders work their dogs.
I'm passionate about this topic! If you would like to contact me privately I would be happy to talk to you about it and help you identify good breeders. There is a fantastic database for known health problems in dog breeds I would be happy to point you toward.