1. Since you're integrating on the surface (a plane), you can substitute z = 4x + y (which is the equation of the plane). This solves your technical infelicity of three variables in the integrand ......
2. It looks like you're using a unit vector normal to the surface (plane). That's incorrect. in general.
In fact, :
3. Are you OK with the region in the xy-plane you integrate over (and hence the integral terminals) .....? The region is the right angle triangle with vertices at (0, 0), (2, 0) and (2, 2).