What is the minimum size to be a natural satellite? Would a smaller satellite be more stable around a smaller planet where the same satellite would not be stable or last around a larger planet?


I added the picture to show how moon size planets can have even smaller moons.
As long as the satellite is way smaller than the object it is orbiting, the mass does not make any difference to its orbit. 

This is because even while gravitational force depends on product of masses of both objects, the acceleration of orbiting object is this force divided by its mass. Therefore any change to its mass will keep its acceleration and thus velocity and thus orbit about the same as before (as long as there are no significant changes to barycenter the two objects are orbiting; gravitational waves do change things, but only for very extreme cases such as close black hole and/or neutron star binaries)

This will get more complicated if there are more than one satellite orbiting the body, and stability in these cases is often hard to determine (aspects like orbital resonances and phase do matter in these cases)

EDIT: added explanation why mass does not affect orbit