(a) Is this guaranteed to happen, a human universal or is it a contingent feature of our culture?

I believe it's a quite general phenomenon - Japan did it, Russia did it, USA did it, all of Europe did it, etc. It looks like a pretty solid rich=slower-growth phenomenon: http://en.wikipedia.org/wiki/File:Fertility_rate_world_map.PNG

And if there were a rich country which continued to grow, threatening neighbors, there's always nukes & war.