If you restore a backup, you will lose all users and posts since you made the backup, so don't do that.
The applicable permission would be in Global Permissions, Groups permissions, Advanced, Private messages, Can remove private messages from own folder, Yes.
Run the user-based permissions mask on a username to make sure there is not a Never permission for this somewhere.
As for the log files in the ACP, same deal but the permission is in Global permissions, Adminstrators, Advanced, Misc, Can clear logs
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