How do I edit songs that are in the Music app on my iPhone 5? I need to delete some songs that were automatically synced that I don't want.
iOS 7 has changed the face of the Music app somewhat. The biggest thing that was introduced is that now, all of you iTunes store music purchases show up automatically in the Music.app. This allows you to stream those songs without actually storing them on your device (an excellent option).
As a result you have two options:
The following is is for iOS 6
Yes, you can.
To delete single songs in Music.app slide from one direction towards the other on a particular song and select "Delete". You can also use this approach with Playlists, Albums, Artists, etc.
If you seeking to delete or edit playlists you can see those buttons at the top of each playlist.
Keep in mind, unless you make the needed changes on your computer, these deleted songs/artists/albums/playlists with sync back to your iPhone.
Actually, there are methods to only check and sync certain songs to your iOS. After connecting your iPhone to iTunes, uncheck the songs you don't want to transfer, then find and check the "Sync only checked songs and videos" box and "Sync only checked songs and videos".
A better solution I personally prefer, is to use something like imElfin iTransfer, which allows me to choose and transfer songs and other stuff very flexibly.