For the first one I know you apply stokes theorem directly and paratize the ellipsoid using spherical coordinates so you get $x=2\cos(t)\sin(w)$, $y=3 \sin(t)\sin(w)$ and $z=\sqrt{35}\cos(w)$ but the dot product of Fdr is so messy, is there a better way to solve this?
For the second one you could apply divergence theorem but the same thing occurs, it's a mess!
For the first part - yes use Stokes theorem, and remember that you should only integrate over the boundary of the hemisphere. Using you parametrization we have that: \begin{equation} \vec r(t) = \langle2 \cos(t),3\sin(t),0 \rangle \Rightarrow \vec r'(t) =\langle -2 \cos(t), 3\sin(t),0\rangle. \end{equation} Thus $\vec F = \langle 10\cos(t)-9\sin(t),0,"nasty"\rangle$ on the boundary. Taking the innerproduct with $\vec r'(t)$ we get that $\int_C\vec F \cdot d\vec r = \int_0^{2\pi}(-20\cos^2(t)+18\cos(t)\sin(t))dt$. Which should be doable.
For the second part use Stokes theorem, and note that the boundary is empty so your integral is zero.