Is Minkowski sum of boundary convex again?

Yes, $\partial C + \partial C$ is convex since it equals $2C$. Equivalently, every point in $z \in C$ is a midpoint of two boundary points. This is obvious if $z \in \partial C$. Otherwise, let $f :S^{n-1} \to \mathbf{R}$ be the continuous function which sends $u$ to the length of the segment going from $z$ to $\partial C$ in direction $u$. Since $n > 1$, this function takes equal values at a pair of antipodal points (a very simple corollary to Borsuk-Ulam, if you want), which gives the desired property.