That's right! You can bring food and liquids via the check point gates at entrance without any problem.
You most certainly can. People bring their own pack lunches all the time. Saves a lot of money if you have several people to feed! One exception - they don't allow you to eat or drink (they allow bottled water) on the tour bus that takes you around the launch pads and over to the Apollo building, but if your food and drink is in a backpack or similar, you'll be fine. Actually, the Apollo building has a lovely waterfront area out back which is a very nice place to take a break, have lunch or a snack, and enjoy the view.
I would honestly think you could. I think they just check for firearms upon entry.
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