We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mc(q)$. Moreover, we show that these two properties still hold on $\mc[q,q^{-1}]$ for the integral version of the quantum graph algebra. We also study the specializations $\Ll_{0,n}^\e$ of the quantum graph algebra at a root of unity $\e$ of odd order, and show that $\Ll_{0,n}^\e$ and its invariant algebra under the quantum group $U_\e(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.