This only concerns (1).

We have that $\Psi_0(G'_{\overline k})$ is the based-root-datum direct sum of $\Psi_0(\eta(G)_{\overline k})$ and the root datum of the almost-direct product of the simple factors of $G'_{\overline k}$ not contained in $\eta(G)_{\overline k}$, 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 $\Psi_0(G_{\overline k})$ as the based-root-datum direct sum of $\Psi_0(N_{\overline k})$ and $\Psi_0(\eta(G)_{\overline k})$, 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 $G_{\overline k}$, then $(\eta_{\overline k}(B), \eta_{\overline k}(T), \eta_{\overline k}(\mathcal X))$ is a pinning of $G'_{\overline k}$, and we compute $\Psi_0(G_{\overline k})$ and $\Psi_0(G'_{\overline k})$ in terms of these pinnings, and then let $\Psi_0(\eta)$ be the obvious map.