Proving that $\iint_S (\nabla \times F) \cdot \hat{n} dS =0$

Yes you can use Stokes theorem but as well you can use Gauss(divergence) theorem

$$\iint_S (\nabla \times \vec{F}) \cdot \hat{n} dS =\iiint_{\text{Interior}(S)} \nabla \cdot ( \nabla \times \vec{F}) dV$$

But divergence of curl is identically zero ie

$$ \nabla \cdot ( \nabla \times \vec{F}) = 0$$