Q: Ipod nano won't add new song when I choose ''Edit playlist''?

First, if you have songs on iPod that are not in your current iTunes library, you're taking a risk.  iPods get lost, or they break, or they have data corruption.  You should off-load songs on iPod and add them to your iTunes, making your iTunes library complete.  It's your "master copy," that you sync to iPod.  If iPod is lost or it breaks, you don't lose your songs; get another iPod (or other device) and take just a few minutes to set it up to sync your songs again.  That's the intended design of the iTunes/iPod system.

And by design, iTunes syncing is generally in one direction only, from iTunes library to iPod.  However, there are third-party methods and utilities that can transfer from iPod to computer.  If you do an Internet search on something like "ipod music transfer," you should get some links, including an article by CNET on the topic.  Once all songs are in your iTunes library, back up your computer's user data, to protect against failure of computer or its drive.

After your iTunes library is complete, set up automatic syncing with your music library, or a selected portion of it.  If your Entire music library is too large to fit on iPod, a convenient way to set this up is to create one or more playlists in your iTunes library, with just the songs you want on iPod.  On iPod's Music settings screen, set it to Sync Music, with Selected playlists, artists, albums, and genres, and select those "iPod playlists" for syncing.

Instead of manually updating songs directly on iPod (a tedious process), you update those "iPod playlists" (add/remove songs) in your iTunes library.  The iPod does not need to be connected.  Updating playlists is easier/faster than updating iPod directly.  The next time you connect iPod (or click Sync if already connected), iTunes automatically updates iPod with all changes since previous sync.  Let iTunes do that work.

For you current problem, you may end up having to do a Restore on iPod.  Data corruption on iPod may be causing problem; a Restore give iPod a "fresh start."  Since a Restore erases iPod, you'll need to off-load its current songs (if they're not all in your current iTunes library).  Therefore, you should set up automatic syncing once your iTunes library is complete, instead of continuing to use the manual method.