When I create one line or multiple lines, copy-mirror this with the 'Linked Mirror' tool and then group the whole thing, I can't move them with the Gripper. The Gripper will move, but the group stays at its place.

Now you say that you should not make both sides of the mirror a group, but only one side. Then it does work indeed. But not all lines in the group are by default mirrored. There is a mix to make one shape.

I would say it should still be possible to move the shape in its whole, where the mirror points are also moved along.

Work around is NOT to work with mirrored lines.