Short answers, pulled from the story.
The British government feared that German access to the Persian Gulf and the Abadan Refinery might fall into enemy hands following Operation Barbarossa. The Axis invasion of the Soviet Union in June 1941 made Britain and the Soviet Union formal Allies seeking to secure the Persian Corridor for Lend-Lease supplies.
Fighting began on the 25th of August 1941 when RAF aircraft entered Iranian airspace and Soviet forces invaded from the north. The Iranian government formally agreed to surrender after a ceasefire on the 30th of August, ending active combat by the 31st of August 1941.
General Dmitry Timofeyevich Kozlov commanded the 44th and 47th Armies of the Transcaucasian Front which invaded from the north mostly from Transcaucasia. The 53rd Army of the Central Asian Military District occupied Iran's northern provinces while using about 1,000 T-26 tanks for combat operations.
Reza Shah announced his abdication as Soviets entered Tehran on the 17th of September 1941 before being arrested and placed into British custody. He was exiled as a British prisoner to British Mauritius for seven months before being sent to South Africa where he died in 1944.
Moscow refused to withdraw citing threats to Soviet security despite the deadline for withdrawal arriving on the 2nd of March 1946 six months after the end of war with Nazi Germany. Soviet troops finally withdrew from Iran proper in May 1946 following an official complaint by Iran to the United Nations Security Council.