*- Can I change the "username" that is displayed in the list of users within a group? I want to put there the Name instead of username.
It uses the formatname field, which outputs based off "Name Format" in CB > Configuration > General. You can change it to Name there, but no you can't change it to Name specifically only in GJ.
*- Can I make "Join" disappear? Or else I want to join the users myself and don't let them exit the group or apply to others. Can it be done?
If the group is Open or Approval type they will be allowed to join. Approval will require your approval before they're actually a member of the group though. Invite type won't be visible at all unless they have joined the group already or been invited to join. Beyond that no you can not control access to the Join button.