The drilling and action center around one platoon. While the platoon did exist in Civil War regiments (as half of a company), in practice it was rarely used. Units quickly lost enough personnel due to illness (if not combat) that companies were the smallest practical unit.
Beginning at 14:28, several Union soldiers wading across the waist-deep river and later climbing out of the river are carrying the U.S. Rifle Model 1903, also known as the Springfield Model 1903. Note especially the soldier who climbs out of the river holding his rifle above his head. The Model 1903 is a bolt-action, magazine-fed rifle that was adopted by the US Army in 1903 and used in World Wars I and II, but not the American Civil War which was fought from 1861-1865.