Hi Megan, welcome to the Disney Parks Moms Panel! Your trip is so close - I bet you're getting pretty excited! Our son was around your daughters age when we first took him to "Meet Mickey" and we had so. Much. FUN. watching him enjoy Walt Disney World. I'm excited that you get to make those types of memories with your own family!
To answer your questions:
Yes, you absolutely may bring a small cooler with you into the parks, as long as it's no larger than 24" long x 15" wide and x 18" high.
Yes, please bring all the snacks that you care to carry with you. (The only stipulation here is no glass containers, with the exception of baby food in glass jars.) I traveled solo with my son right as he hit a very difficult picky-eater stage and I brought so many snacks of what I knew that he would definitely eat that it looked like we were moving in.
And finally, Yes, you may bring any non-alcoholic beverage in to any park, in any bottle, cup, or mug, as long as the container is not made of glass.
I hope these answers help in some of your vacation planning, Megan, and I just know that you and your family are going to enjoy experiencing Walt Disney World Resort through the eyes of your daughter.
~Kristen
PS - A few things my son really enjoyed at this age were playing in the toddler zone of
Storybook Circus, meeting some friendly barn-yard animals at the
Affection Section in Disney's Animal Kingdom, and of course meeting Mickey Mouse. (For sure stop by the location on
Town Square in Magic Kingdom Park for an extra special surprise!)