There are different mechanisms to make eyes move and there are different ways to get the eyes (like floyds) to shut. I think they use a string attached to the tops of the eyes that he can pull down on somehow to get them to close, same with Mahna (I think 2 strings attached to a ring that the fingers go thru).
You don't have to have eyes behind sunglasses. A puppet of mine, Rocky Robin, has on shades and no eyes. Then, there's Ziltch The Monster who wears shades and has eyes. Just depends on what you're going for really.