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

You can also rename it. Click on the little x beside the class name, and then, without clicking anywhere else, write the new name. You mustn't click on anything before writing the new name if you want to keep the styles you have given to it.