^
Apparently it's quite a simple formula. Skip to the relevant section(s):
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The formula is health / health + sickness. At the beginning of the game the default is 30 years.
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The approval rate and life expectance show you the potential nations have for future population growth in their cites, as high numbers in both these categories means that there is plenty of room to added pop points to cities without having unhappy or unhealthy people.
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So having (or trading for) different types of food, smaller cities, health buildings everywhere, and forests + environmentalism contribute to making your country into a Scandinavian welfare utopia. That's...pretty close to reality actually.
Most players play for raw power, however, and population is power. Extra health buildings are also quite low on the build priorities unless the player is looking for a buffer for future growth or potential loss of resources. There's always that next wonder to build, one more modern armor to pump out to fuel the war machine, or just plain wealth or science production when there's not much else to do.
And those forests? Well Archon_Wing demonstrated their most common use in one of his posts above.
As for the play Archon and Kaioshin are having. I'd say you tech and colonize and cooperate with Kaioshin to leave the rest of the world behind in their Medieval squalor. Then you can always choose whether you'd crush riflemen with tanks or aim for the stars. What is Kaioshin doing in that game beside culture warring with the Dutch by the way?