No that's wrong. Death doesn't happen because DNA stops, that's why transplants from dead people work.

Cellular death is quite a different thing from systematic death, which is the breakdown of the metabolic system. This can occur after sudden disruption- loss of blood pressure, the faiulure of the heart to move oxygen and nutrients around etc. - or slowly, perhaps when some regulatory mechanism fails. Of course, when the supplies to the cells fail, cellular death follows, which is why organs for transplant must be removed quickly.

And brain death is another matter still. Brain dead people can be otherwise physiologically perfect.

Yes, DNA codes proteins, and needs good repair mechanisms, of which it has several. And sometimes the mechanism fails, occasionally resulting in runaway reproduction of that cell- you get cancer. As I said earlier, it took a long time to get that good.