No, but it can be the same temperature if the water is spinning fast to keep it from freezing
"Ice water" could either refer to water that has been chilled to the freezing temperature or just a container of water with ice floating in that is close to (but still above) that freezing temperature. Ice can be no warmer than the freezing point but can certainly be colder. If the solid (ice) is in equilibrium with the liquid ("ice water"), the ice will NOT be cooler, but as noted, if they are not really in equilibrium, you would expect the solid (ice) to be colder than the liquid water rather than the other way around.
Assuming a cooler with some quantity of ice less than the total volume of the cooler. Add water, wait a few minutes for heat exchange to stabilize temperature of ice water solution. The pop cans you place in cooler will warm to or cool to and, as long as both ice and water remain in solution, will remain at 32 deg f. So adding water to a cooler of ice and soda will stabilize your soda temperature at 32 deg f. and it probably won't freeze.
cooler
Ice cream feels cooler to our teeth than ice cold water because during the change of state some amount of heat is absorbed which is known as Latent Heat. So in ice no hidden heat is absorbed while in cool water latent heat of fusion or melting is absorbed.
It is not.
The ice melts and the water gets cooler.
cooler
The temperature will be the same.
Heat will naturally flow from a hotter object to a colder object. In this case, from the water (or anything else) surrounding the ice, to the ice. Taking heat away from the water will leave it cooler than it was before.
Ice at 0 degrees is generally better for keeping food and drinks cool in a picnic cooler than water at the same temperature. Ice has a lower density and can absorb more heat as it melts, maintaining a cooler temperature for a longer period. Additionally, ice can create a colder environment within the cooler, helping to keep perishables at a safe temperature. However, using ice packs or frozen water bottles can also provide effective cooling without the mess of melting ice.
You should remove the melted ice water in order to keep the cooler colder. As the ice melts, the water will start to warm up and reduce the overall effectiveness of keeping things cold. Drain the water periodically and add more ice as needed to maintain the temperature inside the cooler.
Ice Cold