Because I've gone ahead with trying to parameterize [tex]S[/tex] directly and learned the hard way that the resulting integral is large and annoying to work with, I'll propose a less direct approach.
Rather than compute the surface integral over [tex]S[/tex] straight away, let's close off the hemisphere with the disk [tex]D[/tex] of radius 9 centered at the origin and coincident with the plane [tex]y=0[/tex]. Then by the divergence theorem, since the region [tex]S\cup D[/tex] is closed, we have
[tex]\displaystyle\iint_{S\cup D}\vec F\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F)\,\mathrm dV[/tex]
where [tex]R[/tex] is the interior of [tex]S\cup D[/tex]. [tex]\vec F[/tex] has divergence
[tex]\nabla\cdot\vec F(x,y,z)=\dfrac{\partial(xz)}{\partial x}+\dfrac{\partial(x)}{\partial y}+\dfrac{\partial(y)}{\partial z}=z[/tex]
so the flux over the closed region is
[tex]\displaystyle\iiint_Rz\,\mathrm dV=\int_0^\pi\int_0^\pi\int_0^9\rho^3\cos\varphi\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=0[/tex]
The total flux over the closed surface is equal to the flux over its component surfaces, so we have
[tex]\displaystyle\iint_{S\cup D}\vec F\cdot\mathrm d\vec S=\iint_S\vec F\cdot\mathrm d\vec S+\iint_D\vec F\cdot\mathrm d\vec S=0[/tex]
[tex]\implies\boxed{\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=-\iint_D\vec F\cdot\mathrm d\vec S}[/tex]
Parameterize [tex]D[/tex] by
[tex]\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec k[/tex]
with [tex]0\le u\le9[/tex] and [tex]0\le v\le2\pi[/tex]. Take the normal vector to [tex]D[/tex] to be
[tex]\vec s_u\times\vec s_v=-u\,\vec\jmath[/tex]
Then the flux of [tex]\vec F[/tex] across [tex]S[/tex] is
[tex]\displaystyle\iint_D\vec F\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^9\vec F(x(u,v),y(u,v),z(u,v))\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\int_0^{2\pi}\int_0^9(u^2\cos v\sin v\,\vec\imath+u\cos v\,\vec\jmath)\cdot(-u\,\vec\jmath)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle-\int_0^{2\pi}\int_0^9u^2\cos v\,\mathrm du\,\mathrm dv=0[/tex]
[tex]\implies\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\boxed{0}[/tex]
