The reaction you describe is what known in chemistry as an equilibrium reaction, symbolised by the double sided arrow:
The equilibrium constant of the reaction, $K$, which is the ratio of the products of the activities of the species on the right and the products of the activities of the species on the left is fairly high which means the reaction tends to run more or less to completion from left to right. In a cell (or battery: multiple cells in series) that results in a flow of electrons from the Zn metal to the Cu electrode, resulting in a net positive EMF.
But by reversing the current by applying the reverse voltage over the electrodes the reaction can also be made to run from right to left. The original amounts of reactants and reaction products can thus be exactly restored, assuming the total amount of charge (current times time) is the same as the charge that was discharged during 'normal' use of the cell.
Internal resistance simply arises from resistance of flow to the charge carriers (electrons and solvated ions, in this case).