Hi there! About the session issue with login/logout, you need to check this forum post here
http://www.joomlaplug.com/component/option,com_fireboard/Itemid,89/func,view/id,2974/catid,12/limit,6/limitstart,6/ and apply that patch to your Joomla 1.0.13 site!
Second, I suggest you set the backup type to TAR in your JoomlaCloner Configuration and also the "Enable direct download link" to Yes from the same area->System tab, or simply download the backup through a ftp client!
Let me know if you have further issues! Regards, Ovidiu