I wouldn't recommend bringing a big cooler due to the large crowds. If its for water it is served complementary at the parks.
Yes you can, I suggest a soft cooler on wheels. No glass and no loose ice, buy the ice and keep the bag in tact. Have fun...
I believe there policy is no hard sided coolers. We took backpacks with cooler/lunchboxes in them and no issues.
Yes you are allowed but you will be pulling it up and down the paths and around guests. I saw some guests who were tired just from pulling the cooler.
Yes, you can.
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