Is it permissible to listen to the Qur’an while playing, and is that preferable to listening to anasheed, and what is the ruling on listening to it out of fear of hearing songs?

Listening to the Quran outside of prayer is not obligatory, but playing while listening is a characteristic with which Allah has condemned the disbelievers, as in His Almighty saying: "No fresh reminder comes to them from their Lord except that they listen to it while they play, with their hearts amused." So, one should not imitate those whom Allah Almighty has condemned. Your fear of listening to songs does not justify playing while listening to the Quran. Rather, strive against yourself not to listen to songs.