Unfortunately they are not roughly equivalent. A solid shell or a ring around a star is not a Dyson sphere. The correct Dyson sphere is composed of many separate orbiting elements. A solid shell or semi-shell is unstable and is bound to collide with the star or just wander away. Yes, the Ringworld is unstable. This was one of Niven's mistakes and misunderstanding of Dyson spheres.
Looking at the orbit of the moon which does figure-8s through the hole in the middle, isn't it conceivable that a sun can orbit that same way? Or rather, the two will orbit each other in a strange non-trivial way.
It would have to be a very small sun tough in order to fit trough the hole and not burn the planet to a crisp. So we arrived at the Discworld, where the elephants carrying the world sometimes have to lift their legs to let the sun pass.
Such a small star would probably have to be kept burning artificially, as the smallest mass still allowing nuclear fusion is around 80 times the mass of Jupiter.
The ringworld has a radius approximately that of the radius of the orbit of earth. A quite large sun could fit easily through such a 'toroidal planet.'
