This only concerns (1).
We have that $\Psi_0(G')$ is the based-root-datum direct sum of $\Psi_0(\eta(G))$ and the root datum of the almost-direct product of the simple factors of $G'$ not contained in $\eta(G)$, so there is no loss of generality in assuming that $\eta$ is surjective. I think you probably want to assume that $\eta$ is a quotient by a smooth, connected, normal subgroup followed by a central isogeny; this is automatic if $\eta$ is separable. When $\eta$ is a quotient by a smooth, connected, normal subgroup $N$ of $G$, we have a similar decomposition of $\Psi_0(G)$ as the based-root-datum direct sum of $\Psi_0(N)$ and $\Psi_0(\eta(G))$, and then $\Psi_0(\eta)$ is the obvious map onto one of the direct summands. Thus it remains to handle the case where $\eta$ is a central isogeny.
In this case, if $(B, T, \mathcal X)$ is a pinning of $G$, then $(\eta(B), \eta(T), \eta(\mathcal X))$ is a pinning of $G'$, and we compute $\Psi_0(G)$ and $\Psi_0(G')$ in terms of these pinnings, and then let $\Psi_0(\eta)$ be the obvious map.