[Lynne]

For backing up your database, these links are from the manual:
Backing Up The Database via SSH/Telnet or
Backing Up The Database via phpMyAdmin

Restoring The Database via SSH/Telnet or
Restoring The Database via phpMyAdmin

If you don't have shell access, some people have also reported success with these scripts:
MySQLDumper
MySQLHotxcopy
Bigdump

Regarding the users sending a message and not having any indication it was sent, do you mean they are sending a PM to someone? They should have a copy of the PM in their Sent Items box (assuming they have that option set to Yes to automatically keep copies of the PMs). You may want to consider setting up a Forum and set it so the usergroup cannot see Others Threads. Then, they can just post a new thread (instead of sending a PM) and they can see the thread there (but can't see others users threads).

If you don't want Unregistered users to see content, then in Usergroup Manager, set the Unregistered Users to not be able to see Topics or see the Forums or whatever permissions you want to change for them.