It seems you can on Android. Just close the video and start playback from the drop-down notification. I get audio only but I don't remember if I did that before I started paying for YT.
No. This is one of the main features of the Premium YouTube
But that's dumb. I don't even pay for data. I'm helping _them_.
Not really, they can't show you their ads if you only listen and it's a feature people are willing to pay for.