The Union Army of the Tennessee was named after the Tennessee River, not the state. Thus the word 'the' differentiates it from the Confederate Army of Tennessee, which was named after the state.
The 'Battle Above the Clouds' was the poetic name given to the Battle of Lookout Mountain, fought in Chattanooga, Tennessee, November of 1863.
The Union Navy had strong cadres of foreign-born soldiers. Admiral David Farragut's flagship, the 'Hartford', had twenty-five nationalities aboard at one time; almost two-thirds of the ship's crew were born outside of the United States. Irish and British men made up most of the crew.
George Crook, the famous Native American fighter, commanded the Army of West Virginia and used many unconventional warfare techniques he learned on the frontier while fighting against the Confederates.