Yes! The cart shop is open all year!
Sure can. You can rent all year. If you're renting though, many times the owner will provide a cart https://www.carycartco.com/beach-rentals/reserve-cart/
Yes you can because I was there in December and we thought it would be cool just to go for a walk around the island. After we spent our day there we thought how stupid we were because we probably would have spent a better day on a golf cart.
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