Is the correct zeroth-order wave function $\phi^{(0)}$ an eigenfunction of the perturbation Hamiltonian $H'$?
No, writing $\hat{H}'\psi_3^{(0)} = E_3^{(1)} \psi_3^{(0)}$ will be incorrect. I will give you two explanations for the same, one intuitive and other mathematical.

Intuitive Explanation:
Assuming that you can write $\hat{H}'\psi_3^{(0)} = E_3^{(1)} \psi_3^{(0)}$. But, $\hat{H}_0\psi_3^{(0)} = E_3^{(0)} \psi_3^{(0)}$, and in an overall sense, you can write
$$\hat{H}\psi_3^{(0)} = (E_3^{(0)}+E_3^{(1)}) \psi_3^{(0)}$$
This essentially means that electron-electron repulsion is keeping the wavefunction same as that of single electron Hamiltonian. Now, since in single electron Hamiltonian, each electron does not &quot;see&quot; the other electron, and they are allowed to be in the same spatial position. Now, on bringing the $1/r_{12}$ term inside the Hamiltonian, the energy goes to $\infty$ for such a case, and hence a singularity is created in the system.
So, it means that the $\psi_3^{(0)}$ is not an eigenvector of total Hamiltonian, $\hat{H}$. Since $\psi_3^{(0)}$ is defined to be eigenvector of $\hat{H}_0$, so the initial assumption that $\hat{H}'\psi_3^{(0)} = E_3^{(1)} \psi_3^{(0)}$ is incorrect.

Mathematical Explanation:
You have given $\left\langle \psi_{m \neq 3}^{(0)} \middle| \hat{H}' \middle| \psi_3^{(0)} \right\rangle = 0$, which is correct. Along with that, we know  $\left\langle \psi_{m}^{(0)} \middle|  \psi_n^{(0)} \right\rangle = \delta_{mn}$. However, you could write the given statement $\hat{H}'\psi_3^{(0)} = E_3^{(1)} \psi_3^{(0)}$ provided $\psi_n^{(0)}$ would be a complete basis set (for explanation, see below). However, $\psi_n^{(0)}$ is not a complete basis set, since for a He atoms, there are infinite number of ways to put two electrons in an orbital, since there are infinite number of orbitals. Since you are treating the orbitals as hydrogenic, it is not guaranteed that the orbitals will be orthogonal to each other once the $1/r_{12}$ term is pulled in.
Even for a hypothetical case if we consider only 1s and 2s orbitals are present and are orthogonal to other orbitals even after the presence of the $1/r_{12}$ term, still they would not be a complete basis set. The reason is simple: You have considered only $1s^12s^1$ configurations, and not $1s^2$ and $2s^2$.

For complete basis, proof:
Assuming $\psi_n$ is an othronormal and complete basis set of dimension $M$, one can
write: $$\hat{H}'\psi_3^{(0)}=\sum_{i=1}^M c_i \psi_i^{(0)}$$ Now,
$$\hat{H}'\psi_3^{(0)} = E_3^{(1)} \psi_3^{(0)} \Rightarrow C_m=0
&gt; \forall m\neq 3$$ $$\Rightarrow \hat{H}'\psi_3^{(0)}= c_3
&gt; \psi_3^{(0)}$$
However, this would have been true if, along with other conditions,
$\psi_n^{(0)}$ would have been a complete basis set.


So, from the given information, it is impossible to conclude that $\hat{H}'\psi_3^{(0)} = E_3^{(1)} \psi_3^{(0)}$ mathematically, and from a chemistry standpoint, $\hat{H}'\psi_3^{(0)} \neq E_3^{(1)} \psi_3^{(0)}$.