Do undissolved salts in saturated solvents enter equilibrium with the dissolved salts? For instance, if there is cup of water saturated with a salt to the point where any more added salt will just float around in the solution, are those undissolved salt molecules in equilibrium with the dissolved ones? For instance, do the undissolved ones become partially dissolved for a moment, while the dissolved ones swap places with them? 

Or does this behave in such a manner that whatever salts were dissolved first to reach saturation are the ones that will stay dissolved, in spite of other added salts?
Yes, they enter into a constant equilibrium. Just as we write equilibrium expressions for insoluble salts as equilibrium expressions, like $\ce{CaF2} \leftrightharpoons \ce{Ca}^{2+} + \ce{2F-}$, when you reach the saturation point, solute particles are in dynamic equilibrium between aqueous and solid phase.