There are bathrooms in the conservatory which is centrally located in the trails. As far as footwear, comfort is key. Some of the trails are sand and great for going barefoot while others are crushed coral and can be sharp.
Yes there are bathrooms in the office/museum building that sits right across the street from the trail. Water fountain at same building. For foot wear I was barefoot that was painful then flip flops still not enough. I ended up putting on my kayaking /water shoes. Those were perfect. Hard soul on shoe because those rocks can hurt .
Its really windy and the trails i would sneakers but if you are there more so to see the blowing rocks i would recommend long pants and a jacket along with shoes you don't mind getting sand in. As to the bathrooms i didn't find any on-site
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