Can Joomla! 1.5 operate with PHP Safe Mode On?

Yes it can! This is a significant security improvement.

The safe mode limits PHP to be able to perfom actions only on files/folders who's owner is the same as PHP is currently using (this is usually 'apache'). As files normally are created either by the Joomla! application or by FTP access, the combination of PHP file actions and the FTP Layer allows Joomla! to operate in PHP Safe Mode.