Yes they do...more so than I thought, now that I look at it. I thought global layers automatically appeared on every layout; didn't realize they were 'opt-in'. Now that I look at them, they basically work like 'include' for layers.
Nonetheless, copy/paste would be handy for cases where you wanted to copy 'template' layers and iterate on them in the editor, without having to resort to events to reposition objects and tweak attributes.
On globals: I still wish the editor handled layer conflicts better. You have to be careful you don't duplicate the name of another layer on another layout, because the editor doesn't rename anything or warn you, it just overrides the non-global layer. For a game with many layouts, this means breaking out notepad++ and trawling for layer names.
Oh, and when I was playing w/ global's, I think I spotted a glitch: If you have a non-global layer on one layout, override it w/ a global layer on another layout, then DELETE the global layer, the local layer still says 'Global, overridden', and prevents you from placing objects on it. If you save the session and reload it, this corrects itself. Can you see this, or should I file a bug report proper?