Ok, I'm really confused now.
From the EA Help File under:
The UML Language
UML Diagrams
Structural Diagrams
Composite Structure Diagram
Properties
A property is a nested structure within a classifier, which is usually a class or an interface. The contained structure reflects instances and relationships reflected within the containing classifier. Properties can have multiplicity.
To demonstrate properties, consider the following diagram, which demonstrates some properties belonging to the library class.
There are two parts, libBooks and records, which are instances corresponding to the classes Books and Computer respectively. After dragging parts from the toolbox out to the workspace, right-click on a part and select Advanced | Set Property Type to link to a classifier.

So I create a Composite Structure Diagram and add the class "Library" to it:

Then, so that the classes Books and Computer exist, I create them in a class diagram:

Then, following the help file, I drag Parts from the toolbox onto the workspace:

Next, I set the property types accordingly:

After setting the visibility of the parts to include their type information, I get:

I then set the multiplicity of the libBooks part and create the barcode connector:

Then, to create an instance of the Library, I drag the class onto an object diagram, filling in the dialog box as follows:

And I get this:

After fixing the size, setting the feature visibility to show the types, and adding the connector I get this:

If I want to instantiate another Library, I can drag it from the project browser as before, and have to fix the formatting, visibility, and recreate the connector.
If I instead select everything, hit copy, then paste as new, it creates the new object and parts but does not create the connector/link or whatever it's called. Isn't there some way of creating instances of configured objects? Why can't connectors/links be replicated like the objects/parts are?
If I drag the class Books or Computer from the Project Browser another class "NewLibrary", it asks if I want them to be ports or parts. If I drag the class onto the general diagram space as an instance, then drag it into the composite class "New Library", it shows up all nice and underlined, but when I then drag the "New Library" class onto the object diagram, the internal classes are no longer present. They don't show up on the list of embedded elements regardless of whether or not "Show Owned/Inherited" is checked.
Is this just fundamentally a wrong approach? It seems to follow the help file example. Is there a better way?