Life or Lives

this one is complicated. It depends on whether you are using the plural or the collective sense of either word.

the volkswagon is the car of the people, the people's car.

It would also be correct to say "volkswagons are the cars of the people" but then it loses some of it's punch.

In the case of life or lives of the people, it depends on which way you are using people. Some english words are the same in the plural and singular. deer, for instance, also fish (though fishes is correct, as well).

"the life of the deer" can mean a specific deer or all deer, depending on context. "the lives of the deer" is also correct.

"The life of the people" is the collective social life of the people, as in "The life of the Hopi is still largely communal," or "The life of the medieval peasant was a hard one." If you turn it around and say "medieval peasants had hard lives," it pretty much means the same thing.