How do you set up playlists and your iPod to only update "checked songs"?
With the iPod connected to iTunes, click the iPod's icon to select it; the iPod's Sync Settings page will appear in the main part of the iTunes window. In the "Summary" pane, put a checkmark in the box for "Sync only checked songs" (or something like that), then click the Apply button.
At this point, only checkmarked tracks will sync to the iPod. Note that this setting does not prevent unchecked tracks from being added to playlists; it simply prevents unchecked tracks -- regardless of their source -- from being synced. Another note: unchecked tracks will not automatically play in iTunes; that is, if the second track in an iTunes playlist is unchecked, iTunes will play the playlist's first track and then skip over the second track and start playing the third track.
Unless you have a specific need for this feature, you're p'bly better off not using it. Syncing only checked tracks is an old, legacy setting from the days when iTunes wasn't smart enough to be able to sync selected playlists to iPods. These days, it's much easier to go to the Music pane of the Sync Settings page and set the iPod to only sync selected playlists.