Is it really true that this is not known? Although I only claimed correlation (and am thus surprised that I was downvoted twice, as that claim is obviously true), based on the famous "robustness" of this observation, I strongly suspect that confounding factors like those you mention have already been analysed to death, and found not to eliminate the explanatory power of women's education.
At least, checking these confounders seems an obviously valuable and interesting avenue to explore. If it hasn't been done yet, I wonder what social scientists are doing instead.
I doubt it, because the xUSSR is an obvious counterexample.
Access to condoms is probably a bigger factor.
Unfamiliar with USSR -- is the birth rate high there despite lots of educated women?