How does a liquid melt a hotter solid?
The cold salt water is going to take heat out of the ice cube just as you say, (and so the salt water will be heated a little to an extent depending on their relative amounts  etc.) The ice surface in contact with the salt water will dissolve as the freezing point of the salt water is lower than that of the ice. Eventually all the ice  will dissolve, but this will have taken some small amount of energy to do as explained next. (I assume little ice and lots of solution but whether all the ice will in fact melt will depend on the exact composition and could be found from a temperature vs. mole fraction plot for a solid liquid equilibrium at constant pressure. There could be solid salt-ice &amp; ice, solution &amp; ice, solution &amp; solid salt-ice or just solution.)

Dissolving a solid in a liquid, say salt in water, can cause a drop in temperature (provided the container is insulated) because it takes  energy to dissolve compounds and this can only come from the liquid. (The ions or molecules in the solid must be attracted to one another, by ionic or van der Waals forces, and so to be separated takes energy). The measure is called the heat of fusion (or melting), or enthalpy of fusion which is a positive quantity, for common salt it is $\approx 30 \pu{~kJ mol^{-1}}$ and of ice $\approx 6 \pu{~kJ mol^{-1}}$.