I am a bit confused by this image, because I thought that direction of the electric field would be parallel tot the direction of the current. However, in this image the direction of the current is perpendicular to the electric field. How is this possible?
Hello Lotte!
That is exactly what the Hall Effect is about 
As electrons enter the material, they change direction as a result of the magnetic field perpendicular to the 2D material because electrons are charged particles. Then, because the electrons change direction, there is a Voltage difference along the y-direction (which is related to E_H).
I agree with you that the picture is confusing, but let’s think of the material as if it was in the x-y two-dimensional plane. I think a thinner picture along z-direction would have done the job better.
Is the idea clearer now?
Also, at question 1.1 from the drude model they ask you to determine an expression for R_(xy) as a function of the electric field. They provide you with a hint:
But if the electric field would be depicted as above, wouldn’t that make the dot product between the electric field and dl zero? Because the two vectors are perpendicular or is dl in this case dw?
We want to calculate the voltage drop along the path described by \Gamma and caused by the electric field, so in this case we need to consider the y direction for the path and the electric field.