Architecture

That's a great question and I'll do my best to answer it.

There are a few ways of using mycelium for constructions. Making a brick from mycelium is quite easy to do, even at home. These mycelium bricks at the end of the process they are heated to the point there is no living organism left in them. So I suppose for this process the answer is pretty straight forward.

I really don't know what BIG will present but from what I see in the photo in this article reminded me of the Growing Pavilion. And since 3D printed mycelium is mentioned (see an example here), I tend to assume that it would be a living product. So for this process and in relation to your question about air quality, I could imagine that even tho molds and mushrooms are both in the fungi kingdom, they don't affect humans the same way, but I can't say I'm sure.

Edit: Just found these 7 videos in a playlist, and I wouldn't be surprised if the answer is there:

Mycology for Architecture