Short answers, pulled from the story.
The division of Korea was established on the 2nd of September 1945, when the United States and Soviet Union agreed to split the Korean peninsula into separate occupation zones along the 38th parallel. Two young American officers, Dean Rusk and Charles Bonesteel, drew the dividing line on short notice using a National Geographic map. The arrangement was intended as a temporary trusteeship, but Cold War tensions and failed negotiations made it permanent.
Dean Rusk and Charles Bonesteel, two young American officers, drew the 38th parallel as the dividing line between U.S. and Soviet occupation zones in Korea on the 10th of August 1945. They chose the line because it placed the capital Seoul under American control and divided the country approximately in half. No experts on Korea were consulted.
The Democratic People's Republic of Korea was established in the north on the 10th of July 1948, when the People's Assembly voted to adopt a constitution; Kim Il Sung was appointed prime minister on the 9th of September 1948. The Republic of Korea was formally declared in the south on the 15th of August 1948, with Syngman Rhee as its first president.
The Korean Demilitarized Zone is a 4-kilometer-wide buffer zone created by the Korean Armistice Agreement signed after three years of war between 1950 and 1953. The DMZ follows the military line of contact at the war's end, crossing the 38th parallel diagonally, and has divided North and South Korea since 1953.
The trusteeship plan, agreed at the Moscow Conference in December 1945, called for a four-power administration of Korea for up to five years. The US-Soviet Joint Commission met in 1946 and 1947 but could not reach agreement due to Cold War antagonism, Korean opposition to any trusteeship, and the assassination of Lyuh Woon-hyung, the last senior politician committed to left-right dialogue, on the 19th of July 1947.
In April 1948, Jeju islanders rose up against the looming permanent division of the country. South Korean troops were sent to suppress the rebellion, and the repression escalated after South Korean independence in August 1948. By May 1949 the rebellion was largely defeated; between 25,000 and 30,000 people had been killed and 70% of the island's villages had been burned by South Korean troops.