"Why is it that in hot countries we put salt on ice in the ice cream makers to keep the ice from melting, while in cold countries we put salt on ice to melt it? "
In both cases, the answer is based on the fact that adding salt to an ice water mixture in equilibrium, lowers the freezing point (or melting point) of the equilibrium. When you add just ice to the ice cream maker, the ice absorbs heat from the surrounding and starts melting. At 0C equilibrium is reached and the temperature cannot go any lower. This is not cold enough for making ice cream. When salt is added, the equilibrium will be reached, and kept at the lower temperatures required. But when salt comes into contact with the ice on the road some heat is released because of the solvation process. This melts the ice yielding once again an ice water equilibrium. But because of the presence of salt this mixture will re-freeze at lower temperatures than 0C. In both cases the lowering of the freezing point will depend on the concentration of salt in the mixture.