I'm stuck on part ($a$) of this question. First I let $G(x,y,z) = x^2 + y^2 + z^2 - a^2 = 0$. Also I found the unit normal to be $n = gradG/|gradG|$. Thus using the dot product to evaluate $F\cdot n = z^2/a$. Now how do I find $dS$ and what am I integrating over?
$dS$ is the surface area element. The best way to compute this would be to use spherical coordinates. Since this is the sphere of radius $a$, the surface area element is $dS = a^2 \sin{\theta} d\theta d\phi$. Then you are integrating from $0$ to $2 \pi$ and from $0$ to $\pi$.