Can I bring a cooler with ice

Yes!!

yes. plus food choices are not so good here so i definitely would take your own food.

You are allowed small coolers that you can carry. Larger ones I do not think are allowed.

No hard coolers...over the shoulder cool bags, no glass containers either...

Not a hard cooler but the plastic coolers the bookbag style I saw

No and yed

Yep

Yes you can, but I think it has to be a soft case not a hard case one ... And you can't bring glass bottles