You are not alone in your confusion! What I've done so far:

1) Run the SQL statement as suggested by Clyde (see post #745)
2) Deleted file ADMIN/includes/boxes/image_handler_tools_dhtml.php as suggested by Clyde (see post #745)
3) Created a new file ADMIN/includes/extra_datafiles/ih3_filenames.php as suggested by dbltoe (see post #746) and corrected by niccol (see post #752)

Contents of the new file:
PHP Code:
<?php
define('BOX_TOOLS_IMAGE_HANDLER', 'Image Handler');
The situation remains as described in the screenshot of post #745. When the links Manager, Admin, Preview and About are clicked, nothing happens.

I wish I was a php programmer to check each file and see what's going on but, unfortunately, I'm not. I don't believe that there are security issues preventing IH to work. It may not meet all security requirements of Zen Cart 1.5 but it should work, even if there were risks which I (and other users) would be prepared to accept.