Who does the pre-copy-edited version of a paper belong to? I am aware that the copyright of the pre-peer review version of a paper belongs to the author. Does the post-peer review but pre-copy-edited version still belong to the author? I thought it may do because at this point the manuscript can still be rejected. 

As I understand it, post copy-edits the manuscript belongs to the journal.
The key here is not actually the history of the manuscript but the submission of a copyright transfer that moves the rights in the manuscript from the author to the journal.  

If you sign a copyright transfer form (or a co-author does on your behalf), then all of the versions of the manuscript transfer in ownership to the journal. Author-friendly journals will then typically grant back a wide variety of rights, including the right to distribute preprints, and there are interesting interactions with preprint servers, academic networking sites, etc. The fact of ownership, however, is changed, and that typically applies to previous versions of the work as well.

Because of this, it is also true that the journal might publish your work but never actually own it!  For instance, the ACM now allows authors to choose between transferring copyright and retaining it but giving the ACM a license to publish.  There are pluses and minuses to both approaches, and it is still very much a question of evolving practice.