This is an extremely long shot, but I had this just today again, but then for a Tridion 2011 SP1 HR2 system, with Content Porter 2009 SP2:
Ensure that in your Tridion_CM database, in the TRUSTEES table, the Everyone group has ID 1 and that any other row in the TRUSTEES table does not have its NAME column value set to "Everyone". It sometimes happens that during an upgrade the Everyone group gets the wrong ID.
Disclaimer 1: I am sure that this is true in the versions described above. It may not be the case for a 2013 system. But especially the "This key is already associated with an element of this collection" error had me stuck for a few hours as well, as logically you'd think that export -or import zip files were off, which is not the case at all.
Disclaimer 2: I don't know whether doing direct manipulation is supported (I suppose not), but I do have the queries here if you want them. I'll post them if you want when you can check the DB.