Proving or disproving hypothesis?
This is more of a logic question than it is a statistics question. I would say that the first point to address would be to define "root cause." Does that mean it is a sufficient condition? A necessary condition? Line of evidence #1 (there exist poor countries with few child labourers) precludes the possibility that child labour is a necessary condition of poverty as clearly poverty can exist without it. #2 reinforces that conclusion (again, poverty existing without child labour, just at a different scale). 

Regarding point of evidence #3, I don't see how that is relevant: whether or not some policy exists compelling parents to act in a certain way doesn't affect the relationship between child labour and poverty. 

As you allude to above, correlation is not causality. In this case, it seems from the correlation you can make equally strong arguments going both directions: child labour causing poverty; poverty causing child labour. You can make a logical case for it built on case studies, but making a statistical case for causality would require digging much further into the data.