How does the ether contribute to planetary motion, and how large are its effects compared with other forces?

The ether is a medium for the forces, not the mover of planets. You cannot transport a force such as electromagnetism or gravity through nothing, because nothing by definition does not exist; there has to be something there, and we call that something the ether. Its importance shows up all through physics: Maxwell's equations gave two different results depending on whether the coil moved around the magnet or the magnet around the coil, which pointed to an absolute space, and the Michelson-Morley experiment expected ether interference and found none, which should have meant the earth does not move until Einstein relativized it away by making the speed of light constant and discarding the ether, only to bring the ether back for general relativity. But the ether does not move the planets. A planet moves because God gave it momentum with an initial push, and by inertia it keeps going in a straight line until the sun's gravity curves it into an orbit. In the geocentric model you add the rotating universe: once God gives the universe its angular momentum, it keeps turning and generates three real inertial forces, the centrifugal, the Coriolis, and the Euler force, and those forces hold every celestial body in place, which is also why the galaxies can rotate about ten times too fast for Newton's law. So gravity and momentum work locally, while the inertial forces of the rotating universe govern the whole; the ether is the medium in which all of it happens, not itself a planetary force.