As far as Topsail goes.... you take your stuff and put it on the tram. The tram is pulled by a truck, so I am sure the driver would allow you to put your stuff in the back of the truck if needed. We bought a bike trailer (one for kids) at Walmart and attached it to our bikes and put our stuff in it and just rode our bikes down to the beach.Yuo can also rent them while there. Don't plan on carrying a large ice chest. But I have seen people rigging up the ice chests with wheels to be towed behind their bike. I don't think you will want to lug a big one across the sand anyway. Also, when you get to the end of the tram trail, you will have to use the wionding boardwalk to get out to the beach. It goes across the dunes.
And yes, there are clean bathrooms at the beach.
Topsail is our favorite of all the campgrounds/parks in that area.