When I copy a previous container, so I don't have to keep recreating a wheel, I need to rename the copy, so it doesn't affect the original. How do I re-name the copy?

Simply add another class name after the first one, then restyle, you can have UpTo 5 class names for any element but that's not really a good thing to have, one or two even 3 are ok,
Or on the copy container, add a unique ID Then set the apply styles to that ID only and make your changes