Some comments on your "Critical analysis"
From your post and link:
In fact, this suggest that violent deaths were not "widespread" as 18 of the 33 clusters reported zero deaths. if 54% of the clusters had no deaths then all the other deaths occurred in 46% of the clusters. If the deaths in those clusters followed a standard distribution most of the deaths would have occurred in less than 15% of the total clusters.
Well yeah, I think violent deaths occurring in 18 out of 33 areas surveyed is widespread. Obviously there are some areas in Iraq where there has been less fighting, or even no fighting - not every town is a target for the US and/or a hideout for the insurgents. This is a non-random event. If someone reported that there were forest fires in 18 out of 33 counties in your area of Colorado, would that mean the study was automatically flawed? Of course not. I don't think standard distribution curves mean much in non-random situations. Your critics are contradicting themselves rather than pointing out a contradiction in the study.
It is probable that many of the deaths, especially the oddly high number of a deaths of children by violence, never actually occurred.
How on earth can they say that??