re: Tablet, I'm no expert (i'm considering a similar set up at the moment , TofuVic) but couldn't you jack the tablet / phone into the Mixer? either the 3.5 jack or lightning / etc?

However, i think you might run into synchronization issues if you're separately piping/casting the video to the TV due to wireless latency. If you used a "splitter" (https://www.amazon.com/3-5mm-Audio-Aluminum-Adapter-MacBook/dp/B00OJY5V3O) to separate the video/audio on the tablet and connect it with wires, I think you lose the benefits of having a tablet, since it'd be fixed in place by the wiring.

To do this well, I think it is getting into what software you decide to use. E.g. If your software has a "server" connected directly to the TV, you could use clients like tablets or phones to remotely administer that server. I'd need to do more research on Karafun, but I don't think it has this capability. (one version of a possible software solution would be Karafun on Apple TV (does not exist) with karafun clients on tablets/phones)