You should be able to activate a pot or frying pan to heat food up if you’re near a fire or other source of heat. Not all foods can be heated up though, and heating food isn’t the same thing as cooking it; it only adds a few extra points of morale gain when you eat whatever you heated up.
Moving whole vehicles between Z-levels is a whole different beast, but if you use the foldable parts mod, you can make small vehicles, like shopping carts or bikes, that fold into an item you can pick up to take between Z-levels. I’ve never actually used one, so I’m not really sure what constraints there are and what you need to do to make it foldable, though. Someone else would probably be able to tell you more.
You seem to be able to heat food even after it is already hot, so while that is a logical argument, if you were able to raise Cooking by heating things, then people would be able to grind it up by just heating the same food item over and over. That’s probably why it wasn’t done.
