Please consider my post with care since I am not a specialist !
What I can say is that I agree with you for what concerns the strip (integral = 0) but not for the 2 circles.
For the bottom circle, you must integrate F.n ds on the surface defined by x²+y² < 1 and z=0. You have F.n=0 because n=-k and F has coordinate 0 on vector k (z=0)
For the top circle, you must integrate F.n ds on the surface defined by x²+y² < 1 and z=3. You have F.n ds = (3(x^2+y^2)^3 / ( [1 + (x^2 + y^2)^4 ]^4)
I hope that I am not wrong ...