How much current and voltage has my resistor + LED I have a very simple circuit: 5v -> LED (5mm, forward voltage of 2 V and max. current of 20mA) -> 1k ohm resistor -> GND.

I want to know how much current is flowing in my circuit. I am using this formula:

U = R x I

Since U = 5, and R = 1000 the current is (according to the formula): 5/1000 = 0.005A.

So far so good. The problem is when I try to check the voltage drop over the resistor and the LED. For calculating the voltage drop in the resistor I would use the same formula where I is my current (0.005A) and R is the resistor 1k ohm. The result is: 5 V. 

Something very weird is happening here cause the voltage drop around my resistor should not be 5 V cause I know there must be another voltage drop around the LED. If 5V is in the resistor it would mean that 0v is around the LED (and it should not light up but it does).

Thank you helping me with this basic calculus.
The problem is that if you're trying to measure the current directly, you're going to get the wrong answer. Whether it's due to burden voltage or the (in)accuracy of the meter, it's a hopeless situation. The 0.005A your meter is showing is actually closer to 3mA in reality, but your meter can't distinguish between the two well enough.

If you need to measure current this small then the only way to do so accurately is to use a very low-value shunt resistor and an amplifier. This is the method the µCurrent uses, and it's essentially the defacto gold standard for measuring current in small circuits.