In addition to the Hopf algebra $O_q(G)$ which you mentioned, there is an important twist-equivalent Hopf algebra $A_q$ introduced by Majid, as the "equivariantized coordinate algebra". In modern terminology, O_q(G) is an algebra in a rather unnatural category: $C^{op}\boxtimes C$, where $C$ is the braided tensor category of locally finite $U_q(g)$-modules. Majid's algebra is the image of $O_q$ under the composition $C^{op}\boxtimes C\to C\boxtimes C \to C$, first applying the braiding on the $C^{op}$ factor, and then applying the functor of tensor product. By construction $A_q$ is equivariant for the adjoint action (hence the name) of $U_q$ on itself. Generally speaking, if you are wanting to quantize something related to the diagonal or adjoint action of $G$, $A_q$ is the one you want. If you are considering rather the one-sided action of some subgroup of $G$, then you want $O_q(G)$.

For $GL_n$, the algebra $A_q$ is sometimes called the reflection equation algebra, because it's defining relations related to affine reflection groups.

The algebra $A_q$ also has a nice interpretation as the CoEnd of the tensor functor $C\boxtimes C\to C$; equivalently, it is a direct limit of $V^*\otimes V$, over all finite dimensional representations $V$ of $U_q(g)$, subject to certain natural relations involving duals.

Finally work of Caldero and Joseph-Letzter exhibits $A_q$ as a certain canonical sub-algebra of $U_q$, so that one can view $U_q$ as degenerating both to $U(g)$ and $O(G)$ at the same time; this can be regarded as a non-commutative Fourier transform.

Regarding the question of quantizing the group algebra of a finite group: one issue is that even infinitessimal deformations are necessarily trivial, since (at least over a field of char zero), the group ring of a finite group is semi-simple and so admits no non-trivial deformations. That said, there are the Hecke algebras which "deform" the group algebras of reflection groups; although these deformations are trivial for generic parameters (are actually isomorphic to the group algebra itself), they are still interesting for many reasons.