We took our 1 yr old daughter with us, didn't plan to, but daycare called and said that she was too upset for mom. She was an absolute angel during the interview, no crying, pooping or other tantrums.
for marriage based cases, you are suppose to provide the Birth certificates of your kids. And yes, kids can be proof of a genuine marriage, but I don't think you need to bring the kid as a proof, just the Birth certificate and photos will do fine.
But in case your kid is like mine who can't be without her mom, bring her over!
If you have a lawyer, you can even ask 'em to babyseat. we did!