Yeah, and the only reason we're not constantly on fire is because the oxidation reaction is too slow to be self-sustaining, and it doesn't get hot enough. Regarding sleep deprivation I don't know a lot about it (you need a neurologist or a biologist, and I'm a chemist), but probably yes, since your life sustaining systems would shut down, causing cell death.

You mostly die because of cell death. Drowning? Lack of oxygen, sad mitochondria, cell death. Smoke? Carbon monoxide prevents your hemoglobin from delivering oxygen to your cell, cell death. Cancer? Your body cannot bring needed materials to your cell (since the cancer is taking it all), cell death. Hunger? No food, no energy for your cells, cell death. Fire? Proteins denaturing, cell death (also some other reactions, but that is already enough to kill your cells). EVERYTHING IS CELL DEATH!