There are small picnic areas throughout the zoo, you can bring your own food in as I've seen people bring in cupcakes and other items for a birthday. You are definitely aloud to bring in other food and drinks just no alcohol.
Yes you can
Yes you can bring your lunch and there is several picnic areas.
Of course you can. There are plenty of spots to eat, with tables, chairs and umbrellas. Pretty scenic amongst the trees too.
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