Mosfet - Voltage sum not adding up
I found strange things when the gate was open... a voltage of 1.6V
  between drain-source, even if nothing is going on at the gate.


The Gate in a MOSFET is very well insulated, so when 'open' it can easily pick up a static charge. Your 'strange' readings may have been caused by the FET getting enough Gate voltage to partially turn on. 

When you added the 1M resistor the Gate would have discharged to 0V and then the FET would have turned off, so its Source voltage should also have dropped to 0V (since it is connected to Ground via a low value resistor).

Another possibility is that you connected a 'test component' which had significant voltage drop at very low current. Most multimeters have an input resistance of ~10M, so if you measured the Drain-Source voltage without a pull down resistor the LED could get enough current to drop 0.2V.