In Civil War movies, "Glory" and "Gettysburg" you see the positioning of the officers fairly well in the some scenes. I'm forgetting the title of the book, it was a manual that included a guide to dressing ranks and overall organization. It was written around the time of the Civil War. Each officer had a specific place on the field. If I remember correctly, Lieutenants would be on the wings near the front. Captains and Majors behind their respective companies. Colonels would stay behind the unit to direct their junior officers. NCOs would stay in a third file to direct the efforts of the men in the front of them.
In battles with large scale frontal assaults, like Cold Harbor or Gettysburg, causalities of officers would be just as high as the enlisted men. What happens in a situation where a company commander is killed and juniors officers? Obviously, higher ranking NCOs would take over, but a single Sgt Major can't control a unit. Would a loss of command rout a unit?
Also, the same text I referenced above stated that Sgts. were tasked with shooting deserters and those who fled the line of battle? I've never heard of this happening, but was this a role taught to Sgts. The same text says that the Sgt didn't carry a shoulder arm but would help reload for the enlisted men, fix weapons, keep order, and if necessary kill his own men.
I believe the book you're thinking about may be Hardee's Rifle and Light Infantry Tactics. The one I have is a reprint of the 1862 publishing. In the manual, the first page has a diagram showing the position of sergeants and NCOs in the rear: the First Sergeant as the final man on the right rear side of the formation, the Second and Third Sergeants standing to the rear of the formation. The Captain "on the right of the company, touching with the left elbow" and First Sergeant directly behind them. Because of this fairly front-line positioning, casualties in officer ranks were not uncommon. For this reason, the other junior officers are at the rear. The First, Second, and Third Lieutenant are actually behind the formation by two paces, so that if the Captain should die they can assume command (the word Lieutenant actually means 'placeholder'). However, if all these men in the rear are pushed forward to assume command positions due to casualties, it leaves less in the rear to keep order and discipline. Enter the NCOs.
With regard to Sergeants carrying weapons, Hardee's states that "Sergeants of Infantry will invariably carry their pieces at a light infantry carry, viz at the Right Shoulder". So they did carry weapons into battle. As for shooting deserting soldiers, I'm not so sure. Their instructions were to "assist officers in maintaining discipline, in keeping men in the ranks, and will enforce obedience, if necessary" . While perhaps they wouldn't fire at deserters in the middle of battle, having an armed NCO directly behind soldiers was probably good for discipline. Hopefully this helped, again the book was Hardee's Rifle and Light Infantry Tactics by Brevet Lt. Hardee.