How do I stop purchased music appearing on my iPhone with iOS7?
I had the same exact problem. Ended up being that the song had not cpletely finished being downloaded to the phone, so it could be deleted. Go into settings and tell it to display all music again. Then go back to each song and click the cloud icon to download again. After each song downloads you will have the ability to swipe and delete, just like all of your other songs. I'm running ios7 too, so I'm sure this is your problem too.