Yes
Yes. That won't be a problem. Just no outside food or beverages, with the exception of water bottles. These can also be refilled inside the park. "Security" was no problem at all. We went in with a stroller and diaper bag loaded with essentials and they never asked any questions.
I saw no bag screening whatsoever... My wife has a huge purse and wasn't searched or questioned, neither were the roller bags that the couple in front of us brought in
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