Yes, you can bring food into the park.
I've seen families bring in coolers of food.
Yes, your bags are checked but I haven't seen anyone give up their food or beverages. Not sure what the Party Line at Disney is but I would guess it's ok.
No. You must buy everything inside the park. You could leave it in a cooler in your car and go back out to it on the monorail or the tram
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