Let Σ be a finite type surface, and G a complex algebraic simple Lie group with Lie algebra g. The quantum moduli algebra of (Σ, G) is a quantization of the ring of functions of XG(Σ), the variety of G-characters of π1(Σ), introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid 90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are Uq(g)-module-algebras associated to graphs on Σ, where Uq(g) is the quantum group corresponding to G. We study the structure of the quantum moduli algebra in the case where Σ is a sphere with n + 1 open disks removed, n ≥ 1, using the graph algebra of the "daisy" graph on Σ to make computations easier. We provide new results that hold for arbitrary G and generic q, and develop the theory in the case where q = , a root of unity of odd order, and G = SL(2, C). In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring O(G n). We extend the quantum coadjoint action of De-Concini-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on O(G n). We show that the set of fixed elements of the quantum coadjoint action in the center together with an additional central element generates the center of the specialization at q = of the quantum moduli algebra obtained for generic q, and we prove that it is a finite extension of C[XG(Σ)] endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra K ζ (Σ) at ζ := i 1/2 with this quantum moduli algebra specialized at q =. This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszynska-Lê on K ζ (Σ).