Exhibits and Pages Disappearing Inexplicably

That is a big mystery. There's really no way to delete content other than hitting the delete button.

Just in case, could you list out all the plugins that you have running? It's a long shot, but maybe one of them is doing something odd.

The changes to the permissions on the files directory is the server settings being changed at some point. Changing those permissions for both installations should be a one-time thing. If it is not -- if the permissions revert back -- then that's something odd happening on the server.

It's hard to imagine what might be causing Omeka to delete content. The best thing I can think of is to try to carefully retrace any steps or actions that might lead up to the disappearance. That, and maybe check the mysql server logs to see if any changes there might have made the database get corrupted.