You could free up C-tab by customizing org-disputed-keys and org-replace-disputed-keys.  Or, maybe even better in your case,  you could overwrite Org's setting of C-tab in a hook, for example

(add-hook 'org-mode-hook
       (define-key 'org-mode-map [(control tab)] nil))

and I use C-c for CUA "copy".

I have no idea how to recover C-c and C-x.  How can you use Emacs if these keys are used for something else?  I guess CUA does this by only using them when the region is active, but this is also not a solution because many commands working on the region are invoked with C-c or C-x.

Incidentally, in my Emacs 23.0.93.1, if I turn on
CUA mode, C-c does not do copy, even though cua-enable-cua-keys is set.