How does the thermostat work? I've had the old mercury switch style thermostats before, but those work by completing a circuit, so I assumed they were dependent on power being on.

Googling tells me it’s “a pilot millivolt system”. I guess the downside of this design is that the pilot* is always lit, consuming a certain amount of fuel. Neat old quick start guide for a similar thermostat: http://nebula.wsimg.com/b2bc334b6de3591406ddd01974f01830?Acc...

* pilot is manual piezoelectric; doesn’t require electric power

How does it blow heat around the house?

Not very well, I lived in an apartment with one of those in the hallway, had to set up a fan to blow warm air to the living room or bedrooms or it got way too cold on cold nights. (which fortunately, were rare where I lived). It'd definitely be better than nothing in an extended power failure, but forced air heat is more comfortable

It’s a wall furnace in the center of the apartment, I guess it blows via convection, outward on both sides of the wall.

These heaters are super common in California, and some larger homes have multiple of them. I assume they’re not as efficient as more modern solutions, but they are certainly reliable and low-maintenance.