Mauno's suggestion to upgrade to 1.6.6 is good, but with the problems you have been reporting, I would not going further than that. If you want to manually upgrade your site to 1.6.6 and you have Cpanel access then the following video may help you. It is old and shows how to use Cpanel to upgrade from 1.6.1 to 1.6.2, but the same process will work going from 1.5 to 1.6.6. The one thing I would suggest you do different is login and rename your current moodle directory to something else FIRST before doing anything else...for example, if it is currently called "moodle" then rename it to "moodleBAK". That will essentailly take it "offline" and prevent you from overwriting it when you extract the new moodle source code. Watch the movie all the way through and then give it a try.
Once you have completed the upgrade, then you can run the migration script to update the database if you like, but you don't need to do this unless you are planning to upgrade past 1.6.
Also, once you do this, then you CAN NOT use fantastico to upgrade this install again. You will need to do upgrades manually in the future.
Note: I probably won't be popping back in here to help solve problems, so use at your own risk.
http://www.kentuckyclassroom.com/moodle_videos/upgrading_moodle/upgrading_moodle.html
Steve
EDIT: One other thought....when you install with fantastico, the moodledata directory is created in public_html. So, if that is inside your old moodle directory, then you will need to move it to the same location in your new (1.6.6) directory. If it's on the same level as your old (1.5) install, then you shouldn't need to do anything with it....unless, of course, you want to move it outside public_html (fantastico doesn't put it there, but that's where it should be), then you will need to move it and edit the corresponding line in your config.php file.