And I am doing the Nature faction game, and so does the improvement in longevity only happen when the people born after you achieve Level 3 Faction?
This is true. The first level grants you food. The second brings the pitcher plant. The third is longevity. All of this is in the info for the tech in the tech screen.
I think though, that getting medicine first, which also helps longevity, would help sooner, and cost less.