I don't know what is $|\psi_k\rangle$ (trying to solve for it), but I do know that it's an eigenvector of $H$, so regardless of its actual form, $H$ operating on it should give its eigenvalue times itself. As for the spectrum of $H$, shouldn't it be the same as $H_0$ in the scattering sector? Nevertheless, even if we don't know the spectrum of $H$, we still know that $|\psi_k\rangle$ is an eigenvector of $H$, so we know the eigenvalue $E_k$ comes out even if we don't know its value. Hence the above solution to the Lipp-Schwing equation has an unknown eigenvalue in it, but is still a solution?

In the equation: $\langle \phi_{k'} |H-H_0|\psi_k \rangle$, the $E_{k'}$ comes from acting on the ket with $H$, and the $E_k$ comes from acting on the bra with $H_0$. In scattering I think it's assumed that $H$ and $H_0$ have the same spectrum, and there is a 1:1 map between the eigenstates $\phi_k \rangle$ of the unperturbed Hamiltonian $H_0$ and the eigenstates (in-states) $|\psi_k \rangle$ of the full Hamiltonian $H$.