WoWW finally this is up. Sorry for the unnecessarily long wait, but I hope you enjoy me answering your questions! Sorry if I trail off quite a bit, I have a tendency to do that :D I'll be recording something new soon!
To listen while using other apps, switch the player to Picture-in-Picture (PiP) during playback — it keeps playing in a small floating window (the screen stays on).
To listen with the screen fully off, in-browser playback stops by YouTube's design. Open the video in the YouTube app to keep listening where background playback is supported (e.g. with YouTube Premium).
Open in YouTube app