To your first question, it depends on how you pose your Hamiltonian. In its usual form H=K+V, where V is the bonding among atoms and/or intermolecular forces, indeed the energies are inferred from spectroscopic studies, as anna pointed out. Identifying the peaks to the associated energy levels (and subsequently inferring other physical parameters of the molecule such as bond length, etc) was a major sub-discipline of Chemical Physics in the earlier years.
So relating to your second question, we first postulate that the world is expressed by (separable) vector spaces and interactions by (Hermitian) operators. Once we do this, we find ourselves only able to measure in the subspace of some eigenvector. The "eigenvalue is energy" part is more of a generalization of the experiments we have done and we now do. In the spectroscopy case, we find the energies first and then infer what the Hamiltonian must look like. So naturally, the Hamiltonians are found so that their eigenvalues match the energies we measure. (Btw, we rarely infer the actual form of the Hamiltonian from spectroscopic studies. We usually tie the data to some simple models and do Taylor expansions of terms.)
@Jan: It does not matter whether we are doing spectroscopy on a population of molecules/atoms or a single one (the latter is NOT a non-existing concept). That is because the energy levels of identical species are the same. As long as the energy levels (ie. the eigenvalues of the Hamiltonian) are the same, there will be dominant transitions which result in the peaks in our spectra.
Further, I would like to point out that it is possible to prepare a very localized wavepacket and do experiments on it, if that alleviate some of your concern about the reality of particle in a box.
Hydrogen atom, on the other hand, does not have much to do with particle in a box physically. It terms out that we like to assume separability in solving differential equations. And in solving the Hamiltonian between a single proton and a single electron, we did exactly that and one component of the solution has similarities with the solution of particle in a box. That's about it.
A better example is quantum dots, where you can sometimes change their colors by manipulating their sizes, because size affects spacings between energy levels! Look it up! It was of much interest when it was first proposed and made. Not sure how it goes now.