Do some families have children that are always in trouble to help develop the parents, or to teach the parents and other siblings things that couldn't be learned any other way, but by a disobedient child?
 

My first answer would be: "Not enough data."  My second, third, and fourth answers might be: "Yes," "No," and "Maybe," respectively.

The idea that the Lord would make someone sin (and consequently jeopardize his or her individual salvation) in order to teach someone else a lesson does not seem to have any scriptural support.  And if you ask about Pharaoh (at the time of Moses) I would recommend that you read those passages again in the Inspired Version of the Bible.

The fact is that some problems are clinical (i.e. emotional, behavioral, or psychological) and a few others may have some spiritual component.  Many might simply be cases of some chemical imbalance in the brain and central nervous system that might cause behavioral dysfunctions.  Others might also be (what everybody assumes) cases of misuse of one's free agency--in the form of rebelliousness, disobedience, etc.

I would venture to say that in those cases where there might be some kind of divine design involved, the parents would know that that was the case through Patriarchal Blessings or some other form of personal revelation.  See, for example, the case of Esau and Jacob, in which Rebekah knew in advance that her sons would be antagonists (Genesis 25:21-23).

Too bad I can't give you a more precise answer at this time.  But if I find something else on the subject I'll let you know.