As long as it is unopened such as a bag of chip or a banana but you can't bring in hot food from eateries.
I don't think so the people in front of us for bag check were not even allowed to bring coffee in.
We were allowed to take a bottle of water with us inside. I don't know about actual food though.
Hi. Don't think so. I have to drink my bottle of water before entering the museum.
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