How do series connected lamp dimmers/electronic switches work?
How does a series connected electronic LED lamp switch operate when it has switched the lamp off and there is no voltage reference. 


Without any clue of what type of switch you have in mind I would have to say that when the switch is off nothing needs to happen. The circuit is dead.


  And how does it work when the lamp is switched on and the voltage drop on the switch is different with different power lamps?


Most dimmers provide a constant voltage at a particular setting. What changes is the current drawn by the load.