Always call before hand to check with management.
Should be fine. I would call as a courtesy in case another party happens to be planned.
I think small things like that are fine. I went for a party once and there were cookies. Maybe give them a call to let them know you will be bringing in a box of cupcakes in case they have a limit on the number or any rules about cleaning up afterwards.
Should be. Attended a birthday party there that had cake pops.
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