Use Stokes's Theorem to show $\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2$

This is an effort to get this question off from the unanswered queue. Also here I use a rather general approach than the classical Kelvin-Stokes theorem.

Stokes theorem reads: $$ \int_{\partial M} \omega = \int_{M} d\omega. $$ Hence $$ \int_C y\,dx + z\,dy +x\,dz = \pm \int_S d(y\,dx + z\,dy +x\,dz) \\ = \pm\int_S dy\wedge dx + dz\wedge dy + dx\wedge dz. $$ For simplicity we choose $S$ to be planar surface bounded by the sphere, not the hemisphere bounded by the plane. On the plane $z = -x-y$, hence above integral is $$ \pm 3\int_S dx\wedge dy.\tag{$\star$} $$ Using the natural parametrization of the plane, this integral can be interpreted as the area of projected area $S$ on to the $xy$-plane (or notice $\star dz = dx\wedge dy$). The intersection of $x+y+z = 0$ with $x^2 + y^2 + z^2 =a^2$, projected onto the $xy$-plane, is an ellipse.

The end points for this projected ellipse are: minor axis end points are projection of $(-1/\sqrt{6},-1/\sqrt{6},2/\sqrt{6} )a$, and $(1/\sqrt{6},1/\sqrt{6},-2/\sqrt{6})a$, achieved when we set $x=y$. We can compute the length of the minor axis is $b = \sqrt{3}a/3$. The major axis end points are achieved by setting $x+y=0$, $(\sqrt{2}/2, -\sqrt{2}/2, 0)a$ and $(\sqrt{2}/2, -\sqrt{2}/2, 0)a$, the length of the major axis is just $a$. Hence the project area is $\sqrt{3}\pi a^2/3$ and plugging back to $(\star)$ yields $$ \int_C y\,dx + z\,dy +x\,dz = \pm \sqrt{3} \pi a^2. $$ The sign depends on $C$ is chosen to be rotated counter-clockwisely or clockwisely with respect to the normal vector to the plane $x+y+z=0$.

To address your own question:

Why I got $\displaystyle \iint_{S} \operatorname{curl} \vec{F} \cdot \vec{n} ~dS = -3 \iint_{A} dA = -3 \pi a^2$ if I use $dS = \sqrt{3}dA$ so that $n\,dS = (1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})\sqrt{dS} = (1,1,1)$?

This is not correct, for that if you use the area element $dS = \sqrt{3} \,dx dy = \sqrt{3} \,dA$, the new $A$ is the projected ellipse on the $xy$-plane, having area $\sqrt{3}\pi a^2/3$ (please see the argument above), not having the same area $\pi a^2$ with $S$.

Standard way using Kelvin-Stokes as Arturo Magidin pointed out in the comments: choosing $C$ rotates counter-clockwisely with respect to the unit vector $n = -(1,1,1)/\sqrt{3}$ normal to the plane $x+y+z=0$: $$ \int_C y\,dx + z\,dy +x\,dz = \int_{S} \nabla \times (y,z,x)\cdot n\,dS \\ = \int_{S} (-1,-1,-1)\cdot (-1,-1,-1)/\sqrt{3} \,dS = \sqrt{3}|S| = \sqrt{3}\pi a^2. $$