Yes. If you bring small children I recommend a stroller as later in the afternoon it can get hot and they may get tired.
Yes, you can bring in any food or drinks as long as they can close shut (no straws)
Yes, and as other have mentioned a stroller or a folding canvas wagon makes this much easier.
Yes
You can bring an ice chess just if it has just bottles
A stroller is the way to go. You can pack water, ice and snacks. If you don't have one, the rent them there.
Yes. You may bring in food and beverages. Glass containers are not welcome but I have never been searched.
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