Lots of people use the bible's instruction to execute children who disobey their parents as an argument to say that the book is not a paragon of morality.
Were disobedient children often killed in that time period? And if so, when did that begin to become unacceptable?
In the early Roman Empire, the eldest male of the household held near absolute authority over his family and slaves, including the power over life and death. So the father had the authority, at least to kill his children for disobedience. http://en.wikipedia.org/wiki/Pater_familias#Children
Were disobedient children often killed in that [biblical era] time period? And if so, when did that begin to become unacceptable?
The answer really is that we don't know, and for the moment can't know. We have very little internal supporting documentation for Mosaic law. We have similar law codes, we have references to the historical context and events in the region, but we don't have information on how the bible's laws were understood and carried out. If we did, it'd be a huge boon for biblical studies people, since it would allow for better dating of the bible. But we don't have that, so there's no answer to your question, sadly.
There is an interesting Chinese law that appeared around the time of one of the great famines (the actual date excapes me). It allowed a family that was starving to sell their children for food.