Can I leave a cooler in a hot car and expect it to keep things cold?

No, not for more than a very short window. A cooler only slows heat transfer, and in a car hitting 130-140°F, even a high-end rotomolded cooler will equalize with the outside temperature. Ice melts fast, and food above 40°F for over two hours is unsafe. If you must leave it, use tons of ice, keep it out of direct sunlight, and crack the windows — but it’s better to just bring the cooler inside.