Are babies free to go in the park? Will they let us bring in a cooler for her milk?

Yes

Yes and yes they are very accommodating

Children 3 and under are free, yes you can bring in a small cooler, you know the kind that you get for your diaper bags for your milk, there is a nursing baby center in Showstreet Palace and then throughout the park at the restrooms there are several changing stations. The restaurants will warm up bottles too if need be. There's also first aid stations throughout the park, they even have a parent swap at a lot of the rides so there's a separate line and that's so one adult stays with the kids so other parents can have a chance to ride the rides if that's something you feel comfortable to do.

Yes babies are free, and I do believe you can bring a cooler in but I would call on that to confirm!

Yes and yes, although they inspect all bags etc going in