On another PC (not the one for Open-AT development) I have:
Will that do?
As you may have noticed, there are many Eclipse “editions”, but any can be then extended with other plug-ins to make them doing what you want.
In the Java EE IDE, I don’t think you can build Eclipse plug-ins; just have a look: do you have a “Plug-in Development” perspective? I think not.
From there, you either install the PDE (Plug-in Development Environment) on it, or redownload a fresh Eclipse classic install.
Just a remark: the Eclipse community has switched yesterday (June 23th) to the new 3.6 (Helios) release. If you want to go further on Developer Studio source code exploration and rebuild, you’ll have to stay on Eclipse 3.5 (Galileo). Hopefully, all legacy versions are available for download on eclipse.org website.
Yes, I’ve noticed that - can’t say I really understand it all, though! 
I think you’re right - how would I check for sure?
You mean I’d need to add that to the existing Java EE IDE installation?
How should I do that?
Presumably that shouldn’t affect the existing Java EE IDE installation?
Go to Window > Open perspective > Other
You get the exhaustive installed perspectives list; if “Plug-in Development” is not in the list, you don’t have it 
In Help > Install New Software, add/select the Eclipse Project Updates site ([u]http://download.eclipse.org/eclipse/updates/3.5[/u])
Then choose the Eclipse SDK feature and install it.
Yes, each Eclipse (or Eclipse based application) installation is standalone and just can’t affect any other install on the system.