Does AV1 just emulate the film grain from the source video, or are people adding/customizing the grain when encoding?

Emulates it, which strikes me as extremely silly. The original grain is in the film itself, it's part of it, it's what the image is made of. Fake grain is just noise. Either denoise it or keep the real grain, I have no interest in fake grain.

Can't you turn it off in the player and get the denoised version?

Probably? Though I don't think those kinds of controls are exposed in most of the players I use (usually set-top box video players). Besides, if I'm dealing with a 20 to 80GB 4k copy of something that originally had grain, I prefer it still be present—if I'm dedicating that kind of space, the point is fidelity. If I'm dealing with something smaller than about 8GB 1080p, I probably don't really care what it looks like aside from preferably not having obvious artifacting, so I guess AV1 would be OK in that case, though all else being equal I'll still pick h265 just to avoid the fake-grain concern entirely.

Why? It's a feature you have to explicitly turn on when encoding, and I've not seen people using it for most stuff.

I can't know if it was used in advance, so favor h265. Which is far more common (and so's h264) so it's not much trouble to avoid AV1.

including digital footage with film grain added in post?

I don't watch a ton of films with noticeable fake grain. Even the ones that do fake it, it's barely perceptible, so I wouldn't care on those—not many of them are emulating the huge, obvious grain of high-sensitivity '60s and '70s film, for instance. A lot of what I watch is from the '90s or earlier, so the grain's real.