Short answers, pulled from the story.
Rodgers and Hammerstein first collaborated on a major Broadway show with Oklahoma! which opened on the 31st of March 1943. Their earlier college collaboration occurred in 1920 and 1921 but did not result in a professional Broadway production until this date.
Oklahoma! was the first musical Rodgers and Hammerstein created together and it opened at the Shubert Theatre in New Haven on the 11th of March 1943. The Broadway debut followed on the 31st of March 1943 at the St. James Theatre.
Rodgers and Hammerstein decided to work together because Lorenz Hart sank into deep alcoholism and became unreliable by the early 1940s. Rodgers approached Hammerstein to ask if he would consider working with him after his previous partnership collapsed.
South Pacific won the Pulitzer Prize for Drama in 1950 and was based on two short stories by James A. Michener from his book Tales of the South Pacific. The show opened on Broadway on the 7th of April 1949 and ran for over five years.
Oscar Hammerstein II died in August 1960 before the film version of The Sound of Music was made. The Sound of Music opened on Broadway at the Lunt-Fontanne Theatre on the 16th of November 1959 and was their last work together.