Wooden cooking layout
The steam released by cooking the foods you mentioned, and boiling water outright, would probably damage the wood. Foil might prevent that, if you seal it completely (using some sort of moisture/heat resistant tape). However, that wouldn't eliminate the fire hazard from cooking in an enclosed wooden box--nothing short of adding a layer of fire-retardant insulation would.

Your best bet is to cook on the floor in the middle of your room (open a window if it gets steamy) and then store your cleaned and cooled implements in the wooden enclosure, provided that your floor is not carpeted. If it is, then I would recommend finding someplace else to cook that isn't a tinderbox.