I am wondering if my solution is correct? Thanks.
(a) On $\mathbb{R}^3$, let $\omega = y dx \wedge dz.$ Compute $d\omega$ and $\int_{S^2}\omega$, where $S^2$ is the unit sphere centered at the origin, oriented as the boundary of the ball.
Revised following Ted's advice: $$\int_{S^2}\omega = \int_{D^3}d\omega= -\int_{D^3}dx \wedge dy \wedge dz = -\operatorname{vol}(D^3) = -\frac{4}{3}\pi.$$
(b) Let $f: F \to S^2$ be a smooth map of degree $n$ from a closed, oriented surface of $F$. Compute $\int_F f^*\omega.$
Added according to Ted's hint.
Degree Formula. Let $f: X \to Y$ be an arbitrary smooth map of two compact, oriented manifolds of dimension $k$, and let $\omega$ be a $k$-form on $Y$. Then $$\int_X f^* \omega = \deg(f) \int_Y \omega.$$
Hence, $$\int_F f^*\omega = \deg (f) \int_{S^2} \omega = -\frac{4n}{3}\pi.$$
In (a), your final integral is over the unit ball, not the sphere. For (b), you need to apply the Degree Theorem. It's immediate.