On the iPhone, can I let YouTube run in the background and play the audio while I use other apps? For example, if there is some live stream, or some videos that I can just listen to, and then I'd like to use other apps such as Gmail or Facebook, can I go to those apps without YouTube interrupting the video?

Currently YouTube stops playing the video when I go to another app.
You'll need to signup for YouTube Premium to have this feature. You can signup here:

https://www.youtube.com/premium

The price is currently 11.99$ per month.

A free method of achieving something similar is to use a non-Safari browser such as Opera Mini, and open youtube.com in that browser. Start the video, exit the browser and the audio stops. Then swipe up from the bottom to access the Control Center, you'll see playback controls in there for the YouTube contents. Press play to keep the audio playing.