Hot cooked food should not be stored in tin lined copper, aluminum, cast iron or carbon steel cookware. The only exception to the steel cookware would be if it is enamel coated. Metals will leach into the food over time. There is some debate about the nickle in stainless steel leaching when food is cooked then stored in it, also. Glass or Pyrex casseroles are ok.
Yes, food should be cooled before storing in the refrigerator or freezer. Temperature moves from hot to cold ... the heat in the pot will move into the cooler environment and raise it's temperature, which means the refrigerator is going to have to work a lot more trying to bring the temp inside the box back down.
Taking a cold pot out of the refrigerator and putting it on a burner to heat it up will not harm the food per se, although it is more prone to scorch on the bottom than if it is brought up to room temp first - but can do a mischief to the pot sometimes. It can cause warping, it the pot has an encapsulated disk on the bottom they can sometimes pop off, enameled cookware can develop cracks or chipping in the enamel. It's better to let the pot come up to room temp or there about before giving it the heat.