It is a 12 volt pump. If you are getting 5.7 volts I would suspect you have a loose ground connection somewhere.
The voltage regulator for a 300e Mercedes is mounted on the rear of the alternator. The regulator is a replaceable item, independent from the alternator.
Motors overheat due to excessive current, not necessarily voltage. Normal voltage can cause a motor to overheat if it is stuck (not spinning). The problem is not usually the voltage, but whatever is causing excessive current flow (usually because the motor is not spinning like it is supposed to).
Inside the alternator
Sounds like you have a wiring problem. A short in the wire going back to your rear light socket..(ref: No voltage at connector!) You're going to have to trace the wire to see where it may be broken at.
12-14 volts
If the switch is what is keeping the load from turning on, then yes, there is a voltage across the open switch.
Has nothing to do with the intensity of the LED, and all to do with the voltage/amperage of thediode, and the voltage of the system it is supposed to be used with.
With normal battery voltage, pull the fuse to it, wait 10 seconds, then plug the fuse back in. The computer, with normal voltage, will "re-boot" properly.
From the alternator to the battery should be 14v
Voltage, frequency, current, impedance, and what the circuit is supposed to do are all important.
Some cars have an external voltage regulator most alternators have internal voltage regulators if your headlights are dim at warm idle tap the gas and they get brighter the voltage regulator is most likely the problem
There is no problem with that voltage. As long as the voltage is plus or minus 5% of the nominal voltage of 115 volts it is considered to be in the 120 volt range.