> Egyptians built pyramids in order that their kings would live forever

Pyramids were built for the after-life. Not as a way to extend their life on Earth. Nobody could verify if it actually worked, so it's mostly a question of faith.

> Stone Age people painted the animals they were going to hunt on rocks so that the pictures would entrap the animals' spirit

That's just speculation from historians. It may be that they are right, but there are also other theories. One other theory could be that it is a way of giving thanks to the animals, and a way to honor their loss. Personally I don't think there is a single reason why they painted animals. People have always had different motivations