Short answers, pulled from the story.
The North African campaign began on the 11th of June 1940 when a small British unit crossed the border from Egypt into Libya and captured Fort Capuzzo. This initial raid marked the start of the Western Desert campaign which would rage across the sands of North Africa for nearly three years.
The German Afrika Korps was dispatched to North Africa in February 1941 under the command of Erwin Rommel to reinforce the Italians and prevent an Axis defeat. Rommel led the forces from El Agheila in March 1941 and later returned to Germany for health reasons in 1943.
The North African campaign ended on the 13th of May 1943 when the Axis forces surrendered yielding over 275,000 prisoners of war. The last Axis force to surrender in North Africa was the 1st Italian Army of General Messe.
The Allies' Ultra programme read much enciphered German message traffic especially that encrypted with the Enigma machine to aid in cutting the Axis supply line to Tunisia. The British also instituted improved call signal procedures and created a fake signals network after the 621st Signals Battalion was overrun during the Battle of Gazala.
Operation Torch started on the 8th of November 1942 and finished on the 16th of November 1942 to secure victory in North Africa while allowing American armed forces to engage against Nazi Germany. The landings met no practical opposition in Algiers and the city was captured on the first day along with the entire Vichy African command.