Is the Hartree-Fock determinant always the minimum energy single Slater determinant solution to a molecular Hamiltonian?
By construction, any determinant $|\chi_{i_1}\ldots \chi_{i_N}\rangle$ where the $\chi_{i_j}$ are distinct solutions of the Hartree-Fock equation $f(x) \chi(x) = \epsilon_i \chi(x)$ will have stationary energy $\langle \chi_{i_1}\ldots \chi_{i_N} | H |\chi_{i_1}\ldots \chi_{i_N}\rangle$ (where stationarity is with respect to perturbing the orbitals).

This is not true: only the converged Hartree-Fock determinant $|\Phi_0\rangle = |\chi_{i_1}\ldots \chi_{i_N}\rangle$ has stationary energy with respect to orbital perturbations.
Stationarity of energy requires all orbitals in the determinant to fulfill the Hartree-Fock equations
$$
\hat f \chi_i = \varepsilon_i \chi_i
$$
where the Fock operator
$$\hat{f} = \hat h + \sum_j^N(\hat{J_j} - \hat{K_j})$$ depends on the $N$ occupied orbitals $\{\chi_j\}$ included in the determinant.
The Hartree-Fock determinant $|\Phi_0\rangle$ fulfills this condition and is therefore stationary with respect to orbital perturbation, because the constituting orbitals are converged (in SCF iterations) solutions to these equations, where the Fock operator is calculated from the $N$ orbitals with lowest energies.
Any other (excited) determinant $|\Psi\rangle$ is generally not stationary with respect to perturbing the orbitals. Since other orbitals are occupied in this determinant, it corresponds to a different Fock operator $\hat {f'}$ where the sum is over the orbitals in the determinant $|\Psi\rangle$ which differs from the set of orbitals included in $|\Phi_0\rangle$.
Using this Fock operator $\hat{f'}$ in subsequent SCF cycles, the total energy of the excited determinant is again minimised by tweaking the orbitals. In this course, I suppose, it is quite likely that the excited determinant relaxes back to the ground state Hartree-Fock determinant  $|\Phi_0\rangle$. The iterations might also converge to a state with higher energy, but one-determinant representations of excited states are rather inaccurate anyway.
In principle, in should be possible for an excited determinant to converge to a determinant with lower energy if and only if the initial converged solution is merely a local minimum. However, this is very unlikely if using a reasonable initial guess.