What type of dog would fit my criteria?

There are no breeds that fit all your criteria. Very few small dogs don't yap; very few small dogs are low-energy, and of the ones that are, almost all of them have long hair. You need to rank these criteria in order of importance. Good for you for rescuing, though!

As for the people who are suggesting a dachshund - not such a great idea. They are HUGE burrowers and diggers. They're also stubborn and bark frequently. (It's a real bark and not a yap, though.) And they chew - A LOT.

The only breeds I can think of that might work are the Pug and the French Bulldog. I'm not very familiar with either but they are two of the very few small, short-haired, low-to-moderate energy dogs that are good with kids. I think they both bark a lot though.

You'd do much better finding a pet from the large and (especially) giant sized breeds with these criteria, than from the small and medium-sized ones. A (short-haired) Saint Bernard would work well, or a Greyhound, or a Mastiff.

Small dogs are almost never good with kids younger than 8-10... because their running around and grabbing hands seem threatening. Big dogs don't have anything to fear from kids, which is why they work better. Plus, big dogs are more likely to be lazy.

If you don't want a yapper then do NOT go with a dachshund. They are by far the yappers of the dog world. A boston terrier wouldn't be a bad choice for you. So many small dogs have long hair, but you could always shave it down short. My favorite toy breed is pekingese. They are laid back, train easily, and get along with cats and other dogs. They are not yappy, but will alert you when a stranger comes along. A pug has short hair and is under 30 pounds, but they are very heavy shedders! You better have your vacuum ready if you want one of those. Plus they are so hyper in the puppy stage. I also love shih tzu's but then again it's the hair thing. But the good thing about the shih tzu's is they don't shed much at all! Mine never shed any! You could try a short-haired chihuahua. Not all chihuahua's are yappy. Just train him/her correctly. Happy hunting!

Your criteria is mostly up to training, but a Coton de Tulear is wonderful! Two of my friends have one and they are both very sweet. They are a pretty good size. They're a sturdy smaller dog.

Boston Terriers are great as well. They seem to be quite small.

Cavalier King Charles Spaniel.

...And there are many more wonderful breeds out there!

French Bulldog

How about a boxer??? Loyal, courageous, gentle, intelligent, athletic, etc... I have a 10 yr old boxer and a puppy mini poodle x lab and they get along great. The boxer is good with about 45-60 minutes of exercise daily + a small yard for easy bathroom breaks.

These breeds should fits most of your requirments. As to the cat, you will have to let the cat sort out the dog, orr teach it about the cat yurself.

Boston Terrier

Pug

Corgie

Poodle (despite the hair)

Maybe some kind mix with French bulldog (freedom dog Boston terrier mix

the first dog breed that came to my mind when i saw this was the pembroke welsh corgi. i have one and hes sooo sweet and cute and never barks unless he want in side when he goes outside. he loves to snuggle with me and he is very friendly.

the alaskan klee kai is a dog that fit your description its not technically short haired like a lab but its medium. its pretty much just a minature husky. very cute though and sweet.

hope i helped

your criteria is going to be difficult, and more than 50% is up to you to train them...

Yorkie- yaps, but all small dogs yap. long hair, but you can cut short, the rest you really have to train. Many are hyper, but it's up to you to train her not to.

Carin Terrier.

Border terrier- I would REALLY recomend these.

I would suggest a dog from the pound or local humane society.

your looking for a basenji. they do not bark at all good with kids meet the weight limit and here is a link to check out the other qualities this breed has. http://www.dogbreedinfo.com/basenji.htm