1, 느라고 is 'because' or 'since'
음악을 듣느라고 벨소리를 못 들었어.
I couldn't hear the bell because/since I was listening to music.
If you say '느라고', the reason or excuse comes before.
And it's used when you say second thing couldn't happened because of first thing.
2, both 으며 and 으면서 is 'during' or 'while'
나는 음악을들으며 소파에 앉아있었어.
I sit on the chair while/during listening to music.
As you know, two things happen at the same time.
3, 으니 is also 'because' and 'since' but different from 1
음악을 들으니 기분이 좋아졌어.
I felt better because/since I was listened to music.
The reason comes before '으니'.
It's used when you say second thing happens because of first thing.