Your final question is very interesting, it gets at the heart of the matter: the Banach-Tarski theorem shows us that in your universe of infinite points and no

Your final question is very interesting, it gets at the heart of the matter: the Banach-Tarski theorem shows us that in your universe of infinite points and no atoms, there does not exist a definition of volume which is consistent. So in this universe the water level in a tub cannot be defined.