There are infinitely many planes that are parallel to any two given vectors. Knowing a point on the plane allows us to find a unique solution.
I don't really understand your argument. In (1), you talk about a plane parallel to the lines given by
(at least I think that's what you mean - I changed to a parametric form to avoid the division by zero). Then you say that there is no such plane that passes through (0,0,4), but this is clearly untrue: The plane meets both conditions.