Your region reg is a circle, not a surface. Your vector n is normal to the circle, but not to the surface. What you are calculating is not a surface integral.
I would do the surface integral like this:
surf = ImplicitRegion[{x^2 + y^2 == 2*z, z <= 2}, {x, y, z}];
n = Normalize[D[x^2 + y^2 - 2*z, {{x, y, z}}]] /. Abs -> Identity;
Integrate[
Dot[Curl[{3*y, -x*z, y*z^2}, {x, y, z}], n], {x, y, z} \[Element]
surf]
The sign of the final result depends on the choice of the orientation of the surface, i.e., on the choice between n and -n.
