You are spot on. I am now convinced that the DOF in my case is the main factor contributing to the images being out of focus. Checked the DOF tables: f2.8 on D7000 at 55 mm has roughly the same DOF as f4 on D800 at 85 mm. I am now not going lower than f4 and the results are much better.
Still wondering about the shutter speed though. As I use my SB700, my settings are f4, 1/160s, iso 400. I am afraid to go below 1/160s and have VR activated as well. I think it is an overkill, VR should make a difference. Will try to go lower as I get more comfortable.
One specific question I have is about VR and AF-ON button combination. Say I am in AF-C mode holding the AF-ON button to track my son's movement. The instant I am ready to shoot, I press the shutter release button, but if I do it fast, VR will not have time to stabilise the picture. If I do it slowly, I am loosing the precious moment, e.g., my son is no longer smiling or he is looking not in the lens and the picture is gone. Is VR usefull in such situation?