It is not mandatory that the same team stay together, although it must be understood that any changes to the team will impact how they work together.

A new Sprint starts immediately after the conclusion of the previous Sprint. Delaying the start of the next Sprint interrupts progress and can reduce the Time-to-Market. Shortening the Sprint length allows the team to respond quicker to changes outside of the Sprint plan and while still allowing delivery of business value. Adding user feedback to the Product Backlog helps the Product Owner with ordering the backlog to maximize the flow of value.
During the Sprint no changes are made that would endanger the Sprint Goal.

Because the Development Team is the ones doing the work, they are best suited to inspect and adapt the progress toward the Sprint Goal. At minimum, this is done every 24 hours through the Daily Scrum. It is the Scrum Master's responsibility to teach the Scrum Team the value and objective of each Scrum event in order for them to optimize the benefits of each event. Because the Daily Scrum is owned by the Development Team, it is up to them on how they should run it but the Scrum Master can provide guidance to ensure it adheres to the Scrum guidelines.

Time-boxes help everyone focus on the work of the Sprint and personally commit to achieving the goals of the Scrum Team.