Horse {they/them}

Puppy Linux

OBS can output any scene (like a desktop, program, etc.) to a virtual webcam which you can then use in anything that supports webcams
Not sure about the audio sharing though, i assume that would depend how the chat program handles webcams

twitch chat is just a (lightly modified) irc server
you can use most irc clients to chat, though it’s a few more steps to connect than using it in browser