Other than taxis and Uber, is there public transportation?
Miami Beach Trolley Bus is the best and it is free! There are 4 popular loops and it is easy to recognise as it has an "Olde Trolley" bus appearance. The drivers are very nice and will usually let you off anywhere on along the route if asked. Regular buses go to Downtown Miami from Collins and Washington.
Not that I could see, there is plenty of moped rentals, bikes and some golf cart like trolleys. Stay on Ocean Blvd and you won't have to walk too far.
Get quick answers from Lummus Park Beach staff and past visitors.