No not on the island itself, but there is a restaurant/tiki bar right there (across parking lot) that we usually stop at for drinks and dinner after the beach.
No they do not. I don't believe you bring alcohol on the island, they check your cooler when you board the shuttle.
Currently I believe they do not. It got damaged by a storm and is not currently in working order and there are no plans to repair it from the knowledge I gathered last time I went to the island.
Thanks! Your answer is awaiting moderation.