First things first. Welcome to the Vortex.
1- That relay with the fuse, is the fuel pump relay. Originally, your car would have had the relay located in the position indicated by the book but VW found they were having an over heating issue with the relay in that location so, they issued a recall to fix the problem, the repair moved the relay above the fuse box connected with the harness seen in the picture below.
2- The relay is sort of a time delayed relay, it turns on the fuel pump immediately when you first turn on the ignition and then the relay checks to see if the engine is turning, by monitoring a tach signal from the ignition coil, if within 2 or 3 second, it hasn't yet detected that the engine is turning, it shuts off the fuel pump and doesn't turn it back on until it detects that the engine turning.
3- I've never heard of an "air switch", it's the tach signal from the ignition coil that keeps the fuel pump on while the engine is running or cranking.
Modified by ABA Scirocco at 6:28 PM 4-28-2010