You can drive to all those places mentioned. About the only place rental cars aren't supposed to go is the Natural Pool, unless you rent a Jeep. Not sure about parking at Eagle beach. If I remember right, there is parking. Eagle Beach is a long beach, so should be able to find parking spot.
Nice restaurants: La Ramblas and Pure Ocean (not beach attire). Smokey Joes has great ribs and more casual. Palm Beach is fun in the evening. Shops and restaurants. There are parking lots
If you get a car, Baby Beach was our favorite. Fun beach bar. There is a charge for restrooms, but if you rent chairs ($10 or $15 each), they give you tickets..
If you are snorkelers, Arashi, Malmok, and Tres Trapi are where the tour boats go, but you can snorkel from shore..
We were there in a timeshare for 10 days last sprng. I think our Avis rental car officce closed at 5? We are on a cruise in March and have a car reserved from Jay's and I think they want the car back by 7pm at the port. The taxi's and uber aren't that expensive if you have to drop your car off early and still want to go to Palm Beach in the evening.