I am worried if when Edgy Eft is officially released at the end of this month, my Dapper Drake will upgrade itself to this new release, or will I need to download the .iso of the new version?

nah, the only thing that the Update Manager does is look for updates for that distributions. If you wanted to upgrade to Etch (though I'm not sure how to do it by apt-get) you would have to change the repos and execute someting like a apt-get dist-upgrade.

To make it short, as long as your repos are for 6.06, it will recieve updates for 6.06 only.

P.S. Ubuntu's Ship-It service will not ship 6.10 cds when it is released (at least for now).

Well, I think I looked now in the right place, because I found the answer. You can see it here.

It shows how to upgrade from each Ubuntu version to the following one. In the case of the page about upgrades from Dapper to Edgy, it warns you that Edgy is beta software, blah, blah, blah, but I think the instructions will be the same after the official release, or if they change, the relevant page will be updated. As for me, I am not planning to update now, but after the official release instead, so I'll go and visit that page again one it comes out.

It even explains how you can do it with the CD in order to save bandwidth. I guess I'll be doing it this way, because I will download it anyway, to give it out to friends, etc. And besides, my bandwidth is awful.