I agree with PP, shelter animals have this knowledge and gratitude that is second to none. However, with an active toddler there are still some precautions you need to take. Don't ever take for granted that the way the dog acts in the shelter will be how she reacts once she is comfortable in your home.
If your family feels very strongly about the dog and there is nothing in the records, if they are available, that shows any red flags, then I'd say go for it. Make sure that if you do bring the dog home and it doesn't work out that either the shelter will take her back or you have another rescue, etc. that can take her immediately.
We don't have cats, but I had a rescue of mine cat tested in a kennel and he did fine. He got to the adopter's home was fine with one of her cats and hated the other to the point that she had no alternative but to return the dog. Dogs can be very similar to humans in that respect, they don't always like everyone. You should find out pretty quickly whether or not the animals are going to get along. Do make sure to introduce them slowly and make sure the dog is introduced to the cats one at a time so that she doesn't feel like she's being ganged up on. That was one of the reasons my potential adopter had a problem, she didn't take it slow. Sometimes it can take a day or two for the animals to get used to the new addition.
As to kids and dogs, make sure that the child and the dog are not left alone, ever. If you've taken down those baby gates, get them back out and create safety zones for your child and the dog. Make sure the dog has an area that is off-limits to the child so that she has a retreat if things get too noisy, etc. My dogs are all crate trained and they will go to their "rooms" on their own when they want to nap, hide a treasure or have just had enough baby and other stuff for the time being. If you catch the child in the crate and you will if she gets the chance, make sure to remove her and enforce that that is the dog's special place. You can also teach boundry training w/out all the gates, but it takes a bit more time and for the short-term, gates will work just fine. Personally, I don't like closed doors because I think those tend to make a dog feel too isolated and could be detrimental to a dog that has come from a shelter situation.
Once you have determined that the dog is going to get along with the child and the cats, then I'd suggest investing in some basic training. I'd highly suggest Greater KC Dog Training Club, I know a couple of their trainers quite well and they know their stuff.
Good luck and thanks for adopting a shelter dog!
