Melting is endothermic in either the + salt case or the -salt case, so it should be a wash in the final consideration of temperature changes.

I’m not sure the endothermicity of Salt dissolution is enough to make a noticeable difference in this scenario outside of a lab, but I guess I could look it up.

There are a lot of goofy answers here. Adding ice will not make the system cooler, or the ice cooler, but it will make the water cooler because it will make the ice melt earlier. However, if your ice is at 0C already, adding salt will not make things cooler because just adding salt can't take energy out of the system. In reality, your ice is likely -10 to -20 C, and so adding salt helps.

I can't think of any reason that adding salt would make it stay cold longer. The duration of maintaining cold should be a function of starting temperature and the characteristics of the cooler.