Almost! you need 'runs'. “the road runs around the island” As soon as I read your question, I thought of the road that runs around Rarotonga. And in your question, possibly: the road runs around the island beside the ocean. Or, you could emphasise the fact that it is close to the ocean with: “the road runs around the outside of the island”

You are looking for a noun form, I think. Your original idea should be "the road running round the island" or "the road that runs round the island". The general term for the road that runs by the sea is "the coast road", you find these on the mainland too. If you want to stress the fact that it goes all the way round the island you could call it the coastal ring road, though for many readers this will remind them of their city's ring road, not a nice thought when you are writing about a quiet island. 'Circular coast road' would do instead.

Yes, the road follows the coastline.

follows the coastline?

If the road goes around the entire island you can say "The road runs around the island." If it only goes along part of the island, you could say "The road runs along the coast."