I'm not an expert for OA database format, but here are a few thoughts:
You need conversion. Based on a recent thread here, the only (?) method is converting a project to a workspace.
Import *.lay and *.dsn into ADS 2012?
How about creating one project with those old designs that you need now (copy the design files into one proejct), and converting that "design collection" project to an ADS 2013 workspace? That way, all your old designs are in one common place.
And because you like file-based copying, here is one additional thought: Each cell in a library has its own directory, and you can move these cell directories between libraries by file system copy & paste, if technology/layers are the same. Not exactly the way Agilent wants you to work, but I've done this a few times and it worked fine.
In general, instead of getting lost with copy & paste of individual circuits, I would use the new library concept to my advantage, and take the time to properly set up one (or few) libraries with old designs for re-use. Indeed, I would not want to have separate workspaces and separate libraries for each ADS 2009 project. But you don#t have to. You can clean up things by moving cells (designs) between libraries, to create an "existing designs" library structure that is easy to navigate. The interim workspaces/libraries used to import from 2009 format are no longer needed then.
Do you want to copy by creating a "snapshot" that can be modified indepedently, or just open an existing design/library for reading?
Assuming that your existing design is already in a ADS 2011/2012/2013 library, you can add that library to your workspace. By default, your workspace has one library that you created along with the workspace, but your workspace can also access additional libraries.
The easiest way to add an existing library (with your existing design inside) is to add the existing library definition file of that other existing workspace. This will look like the attached screenshot, where I added (loaded) two libraries from two example workspaces.
Once you have loaded that library, you can work with that library, and copy cells (designs) between libraries. I think this is what you are looking for. You can also reference cells (designs) from other libraries without copying, if you don't need to modify the design. So this is actually very flexible and powerful, once you understand that libraries can be opened from/shared between different workspaces.