Identifying unknown components in transformerless power supply
By no means an expert in this sort of design, but I think I figured it out...

Notice that all of the loads it says may be used are resistive (at least, primarily e.g. hairdryers are a big resistive element with a relatively small fan). Hence, the circuit doesn't actually have to convert the 220V to 120V, it just has to scale down the total power. Consequently, a half-wave rectifier isn't a bad choice. This circuit is essentially an active half-wave rectifier. It is somewhat apparent from careful inspection, and confirmed by a crude simulation.

I assumed the diode to be a bog-standard 1N4148, because that's what it looks like. Per marcelm's comment, it's more likely to be a diac, although replacing it with a diac model in simulation didn't change anything. I agree with Vladimir that the carbon-like thing is some sort of temperature cutout circuit, although my simulation doesn't seem to confirm that...

Anyway, here's your schematic, redrawn. I had to use an equivalent circuit for the triac because I couldn't find a working triac model. Note that Vbe of Q1 goes to -600V, so either it's a special transistor or the schematic is not quite right.



Here's the voltage across the load. There are still things to adjust in this schematic, but it's close.



While this simulation may not be entirely correct, I think you can get the gist of the circuit: it's not linearly reducing voltage, it's simply switching on the line voltage some of the time. As such, the output is not sinusoidal, hence why it is only specified for use with resistive loads. That is also why it is able to handle 1600W: the triac, conducting the load current, is either on or off, and is therefore quite efficient. The other components are merely for control, and handle very little power; the only component in series with the load is the triac, therefore it's the only thing that needs to be "beefy". The other components need to handle high voltages, yes, but not very much power.