Is there a requirement to fly the plane advertised? In other words, if I book specifically a non-737 800 Max plane, and the airline switches, can I get my money back, or make them fly me on a different airline?

Why did you put the 800 in there? The 737-800 is a separate plane from the 737-MAX, and doesn't suffer from these problems (and is still operating; I just flew one cross-country last week). You might be referring to the 737-MAX 8 (not 800), which is the only variant of the 737-MAX delivered so far.

My mistake.

Interesting question. I have never had this happen, and I think I'll be asking a lot of questions if it does.

It's not something an airline can do hastily, as different aircraft (well, at least those of differing type rating) require different crews to operate. So the airline will have to grab a totally different crew to match their replacement aircraft.

I assume it happens regularly. I had a couple of flights myself on other plans than initially advertised. I always assumed that the airline canceled/delayed other flights because of it (taking plane a and crew a from flight a to do flight b with plane a).

Before the 737 Max got grounded (after the 2nd crash), I remember reports that some airlines allowed passengers to rebook their flights for free (to end up on a different plane). I doubt they will continue doing it once the 737 Max is allowed to fly again. Eg, Southwest is too dependent on it.

When it has happened to me, the crew came with the plane.