There’s a pretty natural curve as societies become educated and more industrialized, you see infant mortality plummet and overall birth rate drop (there’s a collection of different causal reasons for the drop in birth rate). Negative percentages for a while, then (we assume) stable within a margin of error year over year.

What this means is that we should see various societies level off or decline a little before stabilizing. The only area that really looks at the flattening of population growth as a desperate problem is modern capitalist economies, where you need an ever increasing supply of cheap labor to continue pushing the holy line upwards. A differently structured economy would likely not care much about a stable or stably decreasing world population.

Exactly. Basically as countries develop and get more educated people tend to have less babies. According to Wikipedia 48% of countries already have under a replacement birth rate including all of the European Union. Empty Planet by Darrell Bricker discusses the phenomenon.