As you describe them, Livestream's capabilities are impressive, and I'd like to know more. From what I saw as far as I looked on their site, I'm not at all sure how the components hooked up and what the actual on-the-frontline control interface is like.
Their online video demo hit me as superficial, almost to the point of being deceptive. I mean, that sailing footage shown in the video was not lived-switch by someone with a laptop calmly sitting on the deck. On one yacht racing shoot I remember vividly, I was hanging on for dear life and my crew was below decks taking turns puking into the galley sink. The Livestream video made all that look way too easy, and I think they must have glossed over the switching operations too. ;)
I got a sense from the video that the Livestream system itself provides software based switching of visual sources, but probably in addition to the traditional video camera switching probably handled by a traditional video switcher. Or does Livestream handle video switching too?. I'm thinking the software must convert all video and web feed inputs to a common video streaming protocol, and Livestream must control them all and ultimately deliver audio and video channels to a streaming server. Does this describe anything close to what it does? Would love some background on how it's set up.
...luckily I have a sibling who does this professionally and he's offered to help with the production.
That's very lucky, particularly because, as you described him, he does "video editing & production", and those are key skills here. In a live production, the video director, who calls the moves in real time and maybe does his own switching (real time editing) is key... and it takes an amazing presence of mind to handle it all. Some of the work is best divided up among several people, with headsets used to coordinate everybody.
I know zilch about is how I'd mix the two audio sources (one mic on stage, another in the audience)
I don't see the audio as being particularly difficult, but ideally you should have an audio-mixer (ie, a person, listening and running the controls, in addition to the audio mixing hardware/software). Otherwise, you're going to have multiple microphones open, and that's liable to give you some occasionally very odd sound.
The traditional physical analogy of multi-camera shoot is that each input is a separate audio or video "track", and you can switch and/or fade between these, or layer them. In real time, with multiple cameras, you use the time during which a camera isn't "live" to reset it for the next shot.
It's helpful to plan the basic coverage in advance, and then improvise around that. Overall, though, everything has to run like clockwork, particularly your graphics sources, which need to be in careful sequence. Plan out every detail, things like whether the principle speaker is going to read the questions on Twitter aloud.
I myself wouldn't have a fixed camera on the speaker at the podium, but that's me. Lots of people these days are using fixed cameras on lectures... but, IMO, not having a camera operator forces your main shot of the speaker to be way too wide, and the whole presentation becomes a lot duller. I'd sacrifice the hand-held camera onstage to have an operator on the podium camera.
I think you'll see as you run through it that it may be helpful to have separation of tasks... producing, directing, switching, engineering... depending upon complexity. A live production is much more demanding than one where you record the isolated cameras and edit later. The simpler you can make things, the smaller your crew can become.
When you use iPhones as handheld video sources, btw, try to make their angles distinctive enough that you can get away with the extreme quality mismatch you'll have between those and your main tripod cameras. IMO, you're better off using them for audience coverage and for, say, an extremely wide shot, than you are to try to use them for another angle on your speaker on stage. Pay special attention to "screen direction", and try to keep that screen direction consistent. Your brother can tell you more about that.
When is this going to happen, and what part are you going to handle?