You do not need to bring proof and definitely you can bring in your stroller. Just make sure you don't bring you any food because they will throw it away at the entrance
No proof, just the honor system. I suggest an umbrella stroller or a folding wagon/cart as opposed to a full-size stroller. It's easier to get on and off the ride queues and won't be as bulky.
You don't need to bring anything to prove age. Yes you can take your own stroller.
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