On one panel, I'd like to have 4x4 'boards' that are all the same. When I copy the respective 'group' in EAGLE, I am losing the placement of the components with respect to each other in the PCB layout. How can I fix that?
Because each new group you create (by copying), renumbers the parts in the new group, your software will not recognize that the old PCB copper patterns you created for the original parts have any relationship with the new renumbered parts. Net names will also change and basically you can't expect your software to read your mind.
If you want to have multiple versions of the same board you should probably edit the Gerber files or use your PCB fabrication house to do this task.