I'll comment on the last question, with the group $G = {\rm SL}(2,5).$ All $2$-dimensional irreducible complex representations of $G$ are real valued, so all have Schur index dividing $2$ by a standard theorem ( Speiser?). Since $G$ has a quaternion subgroup of order $8$, the Schur index of such a representation is necessarily $2$. Since the ring of integers of each of the fields mentioned in the question is a PID ( by Minkowski for example, though that is overkill) , it suffices to check whether $G$ is isomorphic to a subgroup of ${\rm SL}(2,K)$ for each of the listed fields. However any field over which such a representation can be realised must certainly contain the field generated by its character. This field necessarily contains $\sqrt{5}$ as it contains $\omega + \overline{\omega}$
for $\omega = e^{\frac{2 \pi i}{5}}.$ Hence the only possible choice for the listed $K$ we need to concern ourselves with is $K = \mathbb{Q}[e^{\frac{2 \pi i}{5}}],$ since the other listed fields do not contain $\sqrt{5}.$

Now for this choice of $\omega$, we note that ${\rm SL}(2,K)$ contains a double cover $E$ of a dihedral group of order $10$, which is isomorphic to a Sylow $5$-normalizer of $G,$ namely $E = \langle \left(\begin{array}{clcr}0 & \omega\\-\overline{\omega} & 0\end{array}\right),
\left(\begin{array}{clcr}\omega & 0\\0 & \overline{\omega}\end{array}\right) \rangle.$ Identifying $E$ with a Sylow $5$-normalizer of $G,$ and inducing the character $\mu$ of this representation, we find that ${\rm Ind}_{E}^{G}(\mu)$ has irreducible constituents of degree $2,4$ and $6.$ By the theory of the Schur index and the discussion above we know that the representation affording the $2$-dimensional constituent $\chi$ may be afforded over $K$ ( and hence over the ring of integers of $K$, since that ring of integers is a PID). For the Schur index $m_{K}(\chi)$ divides $2$, but we have exhibited a $K$-representation of $G$ which contains the character $\chi$ with multiplicity one. (Note that $K$ is not the field generated by the character values of $\chi,$ but a degree $2$-extension of that field, so this is consistent with earlier statements).

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