We have always brought in our own drinks. We started doing this with our baby. We packed the baby bottles in the little cooler. Not only do they not have a problem with it - but the front desk said we could get ice from the vendors to pack around the drinks. The ice is made from reclaimed water and as long as you make it clear that you are NOT going to use it IN your drinks they'll give it to you.
We mix our drinks - chilled and frozen - so we have drinks to start with.
Also, I pack some snacks in my backpack - but now that the boys are 3 and 6 - they have to carry some of their own snacks.
I pack snacks that are alike in large zip loc bags so it makes it easier going through bag check.
Pretzels, hard chips, fruit snacks, fruit roll ups are all good.
Anything with chocolate will melt - fun. And anything soft gets smushed.
Hard candies are good.
We always use the plastic bottles. The kids get the straps and bottle tops. It sounds silly but they do make it easier for the kids to carry the bottles around.