In an ideal charged capacitor (with infinitely large parallel plates), the electric field outside the area between the plates is zero.
Will be there any current flowing through the red wire from plate 1 to plate 2 if I attached it just like on the image below?
I think the answer is no. Because the electric field in the points where I attached the wire to the plates is zero, just like it is all along through the wire.
What I'm a bit confused about is the fact that you can actually say that there IS a path connecting those two points (marked with red dots, connected by the wire) with non zero voltage.
Just look at the definition of voltage:
As long as the integral over the path we take (where $E$ is the electric field) from point 1 to 2 isn't zero, the voltage won't be zero = there will be current. Like in this case (forget about the capacitor plates here, just assume the electric field exists in the area between the virtual plates as in this picture):
The integral over the red path won't be zero, so the current should be able to flow between those two points once they are connected with a wire, right? But there's nothing to push the charge from the first to the second point, because the electric field is zero at those points. Could anyone explain that to me?