Is there a way of removing all music downloaded for offline listening, instead of going one-by-one?
Trying to delete everything downloaded onto the iPhone; is there an "erase all" option?
Yes, but you do it in the settings app, not the Music app. Go to Settings->General->Usage->Manage Storage. This will show you a list of all your apps and how much data they are taking up on your iPhone. Tap on Music, at the top is an item that says "All Songs", swipe it to the left and a red Delete button appears. Tap it and viola, all the downloaded music is gone.