Yes you can during their summer season. There are applicable fees... So you will want to stop in at the front desk.
Yes you should be able to in general, boating, fishing, etc... but there is usually groups there that have paid extra for whatever they are doing so you may not be able to just do or participate in whatever you feel like.
I believe there is a 3 night minimum.
I'm not sure. Call to be on the safe side
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