if you're competitive and want low latency/high framerate: you can do all of this with a single cheap USB physical switch (about $2 on ali express), plus DDC to switch monitor inputs automatically

I have this hooked up, so that I push the switch and the monitor inputs automatically switch

I also have a (cheap) physical audio mixer to do the audio, but you can do the same thing with pulseaudio and a line-in cable from one computer to another