Can Dev. Team add Sprint Backlog Item during a Sprint?

Can Dev. Team add Sprint Backlog Item during a Sprint?

Would this change in the forecast of work improve their ability to meet the Sprint Goal?

Also what happens if the Sprint has been set to 4 weeks but all the work is done after 3 weeks.   What do the Scrum Team do for the extra week?

Regardless of how much work has been done, has the Sprint Goal been met?

Can Dev. Team add Sprint Backlog Item during a Sprint?

The Sprint Backlog is entirely owned by the Development Team and only the Development Team can make changes to it. The Sprint Planning produces the initial Sprint Backlog, it's expected that as the Development Team progresses with their work, they will identify new things that need to happen or determine that some things that they thought would be needed are actually not needed. The key thing to watch is the Sprint Goal - changes to the Sprint Backlog should be aligned with the Sprint Goal and not jeopardize it. If the Development Team is beginning to question their ability to meet the Sprint Goal, there should be discussions with the Product Owner, perhaps facilitated by the Scrum Master.

Also what happens if the Sprint has been set to 4 weeks but all the work is done after 3 weeks. What do the Scrum Team do for the extra week?

What does the team think that they should do? There are a huge number of options in this situation. But regardless of what the team chooses to do, they should learn from this to be able to provide better forecasts in the future - a week of time is a pretty big difference between the forecast and actual.