Short answers, pulled from the story.
Robert Gilruth stood at the head of the Space Task Group when it formed on the 5th of November 1958. He managed America's human spaceflight programs with a team that included thirty-seven engineers and eight secretaries known as computers.
The entire organization moved to Houston, Texas by September 1962. The new Manned Spacecraft Center was built on land donated by Rice University and completed in September 1963.
President John F. Kennedy announced the national goal to land a man on the Moon by the end of the decade on the 25th of May 1961. This announcement prompted James E. Webb to secure approval for building a dedicated center.
Thirty-two engineers from Canada joined the group because they lost their jobs when the Avro Arrow project was cancelled by Canadian authorities. These workers brought expertise that helped scale up the early human spaceflight program.
The MSC facility was completed in September 1963 which effectively marked the end of the original Task Group. The term Space Task Group was ambiguously reused later for presidential advisory committees regarding post-Apollo strategies.