Does the wii browser save cookies or history?

Yes, but they are internal. I do believe there is an option to dump them but I have yet to go looking. Try using the option button. I think you can dump them from there. I’ll check again when I’m home.

Cookies are saved but the only way to get rid of them is to clear your system’s memory, which deletes all your favourites along with it.
History I’m not sure about, except for the temporary history that saves the back and forward buttons.

Yes you can delete your cookies without affecting your favorites!

Make sure you’re using the latest version of the Opera Browser (download from the WiiWare area of the Wii Shop Channel), and then there will be an option to delete your cookies in the Settings, accessible from the browser’s main menu.