How can I control friend requests and nicknames for students?

Go to Classes, choose the class at the top and open the Class settings tab. Under What students can do, you can turn these on or off:

• Nicknames in group practice: when off, students show up under their username, and they can't change their nickname.
• Friend requests outside this class: when off, students can only send friend requests to classmates.
• Game invites outside this class: when off, students can only invite classmates to multiplayer games during school hours.

Changes save straight away. All three are on by default.