Quantum Mechanics of a rigid body that has only finitely many possible axes of rotation

I consider the quantum version of dynamics of rigid body motion with intrinsic angular momentum, having moment of inertia $$I$$. Given a classical free system of one free rotating spherical rigid body (i.e. $$I = I_0 \mathbb{1}_3$$) in the Hamiltonian formulation:

$$$$H = \frac{\vec{L}\cdot \vec{L}}{2I_0}$$$$

where $$I$$ is the moment of inertia, the quantised description by canonical quantization is obtained by simply promoting the coordinates in the classical description ($$\vec{\theta}, \vec{L}$$) to operators with properties from the Poisson brakets translated:

$$$$\{L_i, L_j\} = \varepsilon_{ijk}L_k \mapsto [\hat{L}_i, \hat{L}_j] = i\hbar\varepsilon_{ijk}\hat{L}_k \qquad etc.$$$$

Thus the operators $$\hat{L}_k$$ obey the commutation relation of the Lie algebra $$\mathfrak{su(2)}$$. With the postulate that $$\hat{L}_k$$ needs to be an observable, $$\hat{L}_k$$ is Hermitian and there must be a basis of eigenvectors of $$\hat{L}_k$$. One can then construct these as eigenvectors of $$\hat{L}_z$$, with labels $$|j, m\rangle$$. This is just standard quantum mechanics. But now:

1.Question: I had not specified the accessible phase space of the classical spinning top, because I assumed the top can spin in any direction of $$\mathbb{R}^3$$ and with any magnitude $$||\vec{L}||$$. How would the quantum description change if I restrict the classical phase space, to lets say only discrete directions like $$\pm \vec{e}_x,\pm \vec{e}_y,\pm \vec{e}_z$$ (so it only spins in 6 directions) or more generally, any discrete subgroup of $$O(3)$$?

2.Question: If one identifies the the direction of the rotation axis $$\vec{n} \parallel \vec{L}$$ with a matrix $$U \in O(3)$$ that rotates space around $$\vec{n}$$ by magnitatude $$||\vec{L}||$$, one could write just as well $$H = c \cdot \text{Tr} \ U^\dagger U$$, with the Frobenius inner product. A generalisation would be to let $$U\in U(3)$$ etc. If one quantizes now, would the operator $$\hat{U}$$ still be Hermitian?