I think you're making this a lot more work than is necessary. The equation for the y = 3 plane is (drum roll) y = 3! (That's an exclamation point, not a factorial.) If you look carefully, that's the equation you got from your (erroneous) calculations. I say erroneous because you have, throughout, switched your x and z components; for some reason, the fact that you're technically using a left-handed coordinate system did not, in the end, mess up your cross product.

You cannot find the equation of this particular plane in the form z = f(x,y), because the plane is massively multi-valued in z. The analogy is that the equation x = 3, a vertical line in the xy plane, cannot be solved for y.

What you're really asking is how to plot that plane on whatever software you're using. What software are you using?