You won't be able to cut up tires without ruining your knife, the steel belts are pretty tough. Likewise you won't make rubber bands from a tire like you would a tube, the rubber in a tire doesn't stretch that much.
Headliner material won't make a backpack, you probably won't even get it off the headliner in one piece. I'd use the material from a seat back or something though both our vehicles have re-usable shopping bags in then, but then they also have tools and other gear.
battery, wire, light bulb can make a fire starter. Seems most cars still come new with a lighter, instead of tossing it out put it in the glove box for an emergency fire starter.
I bet you could find enough material to make a snare.