No. The fact there is neither tmp nor public folder is worrying as they are required for the image uploading. They are definitely there when I unpack the archive.
Yes. You will have to copy the database information from your current config.inc.php to the one in Lazarus but then, after backing up, simply replace your existing files and visit install.php.
Not sure who installed it for you as they have changed the footer text, possibly to avoid spam. Anyway go to Free Scripts from the menu and download the latest version (click the little download icon bottom right of the entry). Back up your existing files. Now delete the config.inc.php file from the new guestbook script (not your current one as you will need it). Now upload the new files, minus the config.inc.php script, to replace your current files.