Why do women live longer than men?

Because some androgenes (including testosterone) are performance enchancing drugs that increase your competitiveness and risk tolerance in a wide variety of activities but takes a toll on your body in the long run. Castrated men live longer and I think similar lifespans compared to women.

This probably has evolutionary roots, as men have a much higher variability in the number of offsprings they can have, therefore great 'wins' have greater payoffs for them.

At least in Western countries, men die from physical injury at a higher rate throughout life, even controlling for things like job-related risks. Whatever causes that, is probably related to whatever causes the higher rate of various risky chronic behaviours we also see in men more than women, like cigarette smoking. I do not think this is a whole explanation, but it's part.

Mitochondrial DNA is maternally inherited = improvements that benefit men are lost, improvements that benefit women are inherited?

(There’s probably a lot more to it, and this may not be a factor at all)