Welcome to YT. I have a 3 year old son myself, who used to be scared of dogs. He got used to the neighbor's dog first when we moved, then we got our own pug, and then a yorkie. He has grown up with a cat for most of his life, who he was wonderful with. So IMO if a kid has been around small animals his whole life he should be fine. He never pulls on ears or tails, unlike some of the kids I have seen. But all kids are different and only you know how well your children behave and listen to your directions. Any time other children come to my house I do not allow them to hold Brandy, they can sit and pet her under supervision.
If your kids haven't been around small animals and don't know how to treat them right I would choose a much "sturdier" dog. A "bigger" small dog, more than 10 lbs. Pugs and beagles are great family dogs that stay small. Yorkies have very small bones and are so fragile. Also, there is a small possiblity that a yorkie(and other dogs) might try to "nip" at your child if they hurt them. For that reason I would recommend choosing a dog that would tolerate kid's playing better. It sounds like you have lots on your plate with 3 kids. A yorkie is like a another one, they are a high maintence dog compared to others that need daily brushing and I take mine to a professional grommer every 6 weeks. They are also well known for taking over a year to potty train.
Whatever you decide I hope that your family and the new dog ends up really happy. Yorkies are beautiful and great little dogs, but they are also time consuming, fragile, and take a while to housebreak. Just make sure that you put alot of thought into the type of dog that will fit perfectly with your family. You might want to try a breed recommender like this one:
http://sy.adiho.com/ASA/Controller?a...=33&appid=9901 I am glad you joined this site and are doing research. Good luck!