I can't give you the formulas out of my head right now, but i can give you some edge-cases which should give you a sense of whats going on:
First, let's take your ice cubes. I will assume they are just below the melting point with equal temperature throughout the whole cube. I will also assume, they are totally submerged in a large amount of water just above the melting point. So no heat is transferred inside the cube, no heat is transferred through the water and we assume both ice and water are always at the same temperature. In this case, only the surface area transferres heat and the volume plays no role at all.
Second, let's pretend your ice cube is rather big, possibly very cold and not submerged in water, but perfectly isolated in some container and the container is submerged in water. Your ice cube will stay frozen forever, no heat is transferred, and neither volume nor surface play a role.
But whats in between? If theres a small connection between the ice cube and the box? Imagine an ice lollipop in a box of isolation. The ice cube is cold, so is the top of the stick. The bottom of the stick is maybe at room temperature. In between, theres a temperature gradient, with heat flowing from the hot end to the cold end.
- Area, temperature and thermal conductivity of the surface determine, how much heat is transferred.
- Volume and temperature of the object determine, how much heat is needed to bring the object into equilibrium with the surroundings (i.e. to melt the ice cube)
- Volume, shape and thermal conductivity of the object determine, how heat is transferred inside the object and therefore how temperature is distributed, including the surface temperature.