DT:

Neri was trained for those types of shootings - moving distance, etc.. In those days they had a 6 shot 38 caliber revolver - you had to make those shots count at all costs. The 2 shots that Barzini took were fatal shots - spinal column, heart. If you watch the scene, when Barzini get hit, right before he turns around, the 2 shots are grouped very closely - thats not from luck, that is from skill and extensive training. The shooting position that Neri takes is for steading the shot and accuracy - from training and reptition, he is not going to miss. Neri is just that good.

In the scene where Rocco busts into the room with he other button,. if you look at it when the shooting starts the barrel of the other buttons gun is pointing at the floor and stays there while he is shooting.