No, the voltage drop will not be constant. The voltage across the zener will be constant though. The resistor will drop the rest (which should be fairly constant)
The zener diode should be a 12V unit. It will be dissipating 0.3W when no load is attached. If the load is permanently connected, You could use a 400mW zener, But if the load can be disconnected, it would be far safer to use a 1W rated part.
I would use at least a 5W resistor, possibly even a 10W if I had the space (especially if it had no airflow or was at an elevated temperature). It won't reduce the amount of heat generated, but it will kelp spread it around and minimise the temperature buildup in the resistor.
You'll know if you connect the diode the wrong way because you'll get less than 1V across it rather than 12. It won't hurt anything (as long as the resistor is in series).
The zener isn't a perfect regulator. For currents between 3mA and 26mA, the voltage may vary a little.