I don't believe so. You will need to bring your own.
Yes, you can. Right at the beach
Yes you can on the south beach, the one that's about a half mile from the parking lot. Here is the link to the rates for beach rentals: https://www.loverskeyadventures.com/fort-myers-beach-relaxation/ There is a trolly/shuttle that runs every 15 minutes to and from the south beach as well.
Not that i saw, but there is a concessions stand & a public restroom facility on the west side of the Keys. Most cabana vendors are near the condo/hotel areas. We recently rented bikes from Dolphin Inn Resort & biked the mile to the park.
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