I worked at McDonald's when they have McDLT's, the entire thing was styrofoam. Still hard for a cold stuff to truly be kept cold, even in styrofoam, when it is being placed in a heated "hold shelf." All the large sandwiches were styrofoam - Big Mac's, Quarter Pounders, McDLT's, etc.
It could have been ordered without the Cheese. But of course until I worked there I had no clue you could ask for your burger to NOT have onions (or whatever). My mom had always said they can't do that and we have to scrap them off with a straw.