I think JDOM and the code produced by the JCOOK generator could be organized like the FoF framework.
I would keep the "general" files, valid for all the jcook-generated components, inside the JDOM library folder (or any other new one, like CkF cook framework) and then on each component folder, just keep the particular files and eventually the forked files.

I prefer not.

This is interresting question, but your classes can evolve, so it will not affect the others components you use.
It is a good idea, but at this point, I will propose thinks FoF cannot do to FoF directly. If not it would be a framework-on-framework-on-framework. (FoFoF).

And also, since the begining, Cook choice is to run without third part and learn Joomla.
With this classes directory, you can see all missing things in Joomla.

you are the main chef and it's up to you. I agree with you but I also think your framework has a fork feature what allow us to keep only the overrides into the component folders and the main framework somewhere else, so it would also be possible (and make sense) to create a nice CK FoF or FoFoF (if that make sense).
Personally I couldn't take a look on the FoF yet, and how it really works, I just took a look on the code, and I partially like how I think it would work, I see a lot of lines just to write the code at the "state of the art" level but at the end it's not that useful, by the way I hope to take a deep look on it, before I continue to talk about something I don't deeply know.