I highly recommend you not get an Italian Greyhound at this juncture of your life. Physically, they are on the fragile side of the spectrum, so it wouldn't be fair to the dog to be placed in a home with a young child. Your daughter wouldn't hurt the dog on purpose, of course, but young children tend to grab, pull, and bang in their exploration of the world. When she starts walking, and for some time after that, she will be unstable and uncoordinated. (I'm a mom, and I remember those days :-D)
In general, small breeds tend to not be the best pick for a household with young kids, due to the increased risk of physical or psychological damage, but if you have your heart set on a small breed dog, I would recommend a pug. Yes, they shed, but with frequent brushing, the hair left around the house should be minimal. (And, eating dog hair isn't all that dangerous.)
More important than breed is individual personality, since there is so much variation of temperament within each breed. A good option to consider is to visit a nearby rescue organization. Mixed-breed dogs are just as good as purebred, and it can be hard to find a breeder who puts the dogs' health and welfare above their own wants. And a dog rescue is more likely to be honest about a dog's behavior (and more knowledgeable about a dog's potential) than a private party looking to rehome their dog.
Congratulations on becoming parents; enjoy the ride!
Toy Poodle! I have a toy poodle, almost No shedding, almost odorless! Super friendly to all people including kids.
I also wanted the same characteristics on a dog and found her through http://Petfinders.com She had been posted by a rescue group in our area. She is a fantastic dog . Great personality, friendly to my cats, other dogs and people. She seems to be a mix of three different breeds.
Dachshunds are awesome and pretty docile little critters in my experience. Also, my sister has a rat terrier/chihuahua mix, with some training (they tend to be barkers, or yippers to be more specific) she is now a little guardian of my 6 and 8 year old nieces!
Poodles, shihtzu, clhuahua, ...have had each and mine all love children....personally, in your sitation, i would go with a lab.
Shih Tzu! So loving.
Grew up with Cock-a-poos. I'm 57. I married my "Only Big Dogs" loving husband 17 years ago. He thought this breed would be a lap dog. Mine is 16 lbs., non-shedding/hypoallergenic, fantastic with Grandchildren, other dogs, a hiker, adaptable, SMART, trainable. He is in love. So fun! Highly Recommend!