Non-commissioned officers, unless they were staff NCOs (quartermaster-sergeant, commissary sergeant, ordnance sergeant, etc.) most assuredly fought alongside the privates. In fact, they had important battlefield duties, such as squad leaders, file guides (keeping battle lines in proper alignment), etc. They also assumed command if, as often happened with officers leading from the front, the officers were shot down. There was a special NCO duty -- serving as the color guard. Color-corporals and color-sergeants were responsible for carrying and guarding the regiment's flag. Since the colors were always front and center in a charge, the color guard was a highly dangerous duty and required enormous courage. It was considered to be a post of honor, and there was never a shortage of volunteers for this dangerous duty.

Yes, NCOs often became commissioned officers, and almost always made good combat officers.

Unlike most of the world's armies and navies, the American military has a long tradition of encouraging non-commissioned officers to exercise initiative, and be innovative and flexible in combat. From ancient times, the surest way to victory was "decapitation" of the enemy, i.e., killing, capturing or isolating the opposing army's commanders. This effectively ended command and control of the enemy forces and opposition usually ended shortly thereafter.

This, however, has never worked against the American military. American NCOs are trained to take charge if they are isolated from command and control, and to assume command if the commissioned officers are killed. A World War II German officer wrote a book on his experiences against various armies. He said that when a Soviet force was decapitated, the soldiers just milled around like lost sheep. However, when an American force was hit hard, they would just split into smaller units and continue the fight. Our NCOs have always been trained to use initiative and exploit every opportunity on the battlefield.