Honestly, "third world" is a bad word. It comes from the cold war. A lot of countries we refer to as "third world" are really "second world" in this distinction.

I would say you have high birthrates despite low material growth in patriarchal countries, where men have more leverage over women. Whereas in countries struggling with birthrates, women have higher standards. For example, home ownership is a bottleneck. In the USA the supply of homes is artificially constrained by the older property-owning class to boost the value of their investment (but really it will result in a crash due to resulting population changes). In china, you have different shenanigans in the construction industry.

"Third world" immigrant groups have lower standards at first, but after they reach a certain material status in a "first world" country, they have the same birthrate trend as their native peers.

Don't know. By government numbers, china has incredible home ownership numbers. And China is kind of patriarchal, for example, man having affairs is seen as a status marker and more or less socially accepted.