vkj's answer addressed the third part of the question. I'll answer the first part.
Which quantum chemistry methods can be used in an external electrostatic field? Also which software supports that? I'm interested in anything from semiempirical (eg XTB) to DFT (eg ωB97, B3LYP-D3) to coupled cluster (eg DLPNO-CCSD(T)). The only starting point I have is a few mentions of Field= keyword in Gaussian and a mention of an undocumented efield keyword in the ORCA forums.
An external electrostatic field can be implemented in any method, since it's just a trivial modification of the one-electron Hamiltonian as $\hat{H} \to \hat{H} - e\phi({\bf r})$. Most programs, such as GAUSSIAN and ORCA that were already mentioned as well as others like Psi4, only implement static fields, ${\bf E}={\bf E}_0$; you give the cartesian components with the necessary keyword. The electric field is the negative gradient of the potential, ${\bf E} = -\nabla \phi$, so for a static field $\phi({\bf r}) = \phi_0 -{\bf E}_0 \cdot {\bf r} $.
Now, the big conceptual problem arises: the potential diverges to $-\infty$, meaning that no state can be really bound in such a potential. If your basis set is extended enough, there will always be artificial states far away from the system because of the divergent potential. Often, this is not a problem in practice, if the basis set does not reach far enough. Another route is given by perturbation theory, that is, you can build a Taylor series at zero field: $E({\bf E}) = E_0 + \nabla_{\bf E} E \cdot {\bf E} + \frac 1 2 {\bf E} \cdot \nabla_{\bf E} \nabla_{\bf E} E \cdot {\bf E} + \dots $. $\nabla_{\bf E}$ is just the dipole moment, $\nabla_{\bf E} \nabla_{\bf E} E$ is the polarizability tensor, then you have the first hyperpolarizability (rank-3 tensor), and so on.
The most important thing to note, however, is that in order to correctly describe behavior in an external field, you need to have a flexible basis set. Already reproducing the dipole moment correctly requires diffuse functions; it is a textbook example for their necessity. For instance, if I remember correctly, an augmented double-zeta basis set gives you the dipole moment of the water molecule as accurately as a non-augmented quadruple-zeta basis set, at the Hartree-Fock level. I think polarizabilities may be even more demanding to get accurately.
