Safari has issues. Don't use Safari. You'll join the stage and won't be able to talk or you'll talk and some people won't hear you. Everything else should be fine. I join from my desktop and mobile equally. I use Brave and Chrome without issues. Well, I guess the only issue on mobile is you have to keep the browser active otherwise it will disconnect you from the nest.

You can record. When hosting, use the gear icon to record. When, you stop recording, you'll be promoted to download the audio.