In my opinion, applying permissions to anything more granular than lists,
libraries and SharePoint groups will just cause you trouble down the line.
MS best practices for permissions apply just as well to SharePoint as they
do to file systems.

You can copy documents from one document library to another with SharePoint
Designer or you can download a document to your desktop and then upload to
another document library. I very much doubt that the versioning will follow
but it's easy to test.

Or you can go to Site Settings -> Manage Content and Structure, then browse to
your document libraries, klik Actions -> Copy and then specify your destination.

This is a good idea. To make it work you need to turn on the Office
SharePoint Server Publishing on the site  collection as well as the
site from which you want to copy documents.