Downtown Saint Paul is pet friendly and kid friendly. Yes you can live in a condo, even a loft style condo with your children and your pets. There are parks to play in and places to walk to and things to do.
Always check the association rules before buying. I have not seen any rules limiting the number of children in a unit but there are pet limits in most. Some buildings have height and weight restrictions. Some will allow cats but not dogs and there is a condo building near downtown that will not allow cats, dogs, birds, hamsters or any kind of pet other than fish. I think they may even have a limit on the fish but I am not sure it can be easily enforced.
You might want to get your dog a sweater if you plan to live downtown.