Eva foam is the perfect material to put over the metal on the back doors. It insulates, sound deadens, its a closed cell foam and of course it also cushions. Yoga mats are made from it, the interlocking puzzle floor mats are made from it. Quite a few camping pads are made with EVA foam. I insulated the ceiling of my vintage fiberglass trailer with it using the version from auto upholstery sources that is called Landau Foam. It is also used as a thin, decoupling, sound absorbing layer under the click together plank flooring material. But that floor padding is too thin for any decent insulating or cushioning.
So the thickest is going to be the puzzle mats or the really thick exercise/camping pads. The Landau comes in a variety of thicknesses, the yoga mats will likely be a little too thin for a lean against cushion.