Q: iTunes problem with checking/unchecking selection

There is a way to bypass this issue with a more convenient way to manage songs on your iPod.  If you want to try this...

In iTunes, create one or more playlists, and put all the songs you want on the iPod on the playlist(s).  In your case, these songs are the currently checked songs, so you can sort your music library using the checkmark column to get them all together on the song list.  Select all songs with checkmark and drag to the new "iPod songs" playlist (or right-click selection and Add to Playlist -> [name of playlist]).  Or you can use existing playlists.

Then, select the iPod in iTunes (click iPod's device button), to show its settings screen in iTunes window.  Along left side of window (in the sidebar), under Settings, click Music.  To the right, the iPod's Music setting screen is shown.

NOTE:  For anyone else reading this, any songs on iPod that are not in iTunes library (on computer) will  be lost.

At the top of window, Sync Music should already be checked (checkmark it if it's not).  Below that, choose the option to sync Selected playlists, artists, albums, and genres.  Below that, under Playlists, find and select (checkmark) the playlist(s) with your iPod songs.  You don't need to choose anything else on that screen.  When you click Apply, iTunes syncs the songs on the selected playlist(s) to the iPod.  Going forward, instead of using the checkmarks, just update the selected playlist(s) in your iTunes library (add/remove songs).  The iPod does not need to be connected.  The next time you connect the iPod (or click Sync if already connected), iTunes automatically updates the iPod with the same changes.