The S-matrix (scattering matrix) is the unitary operator $S$ that determines the evolution of the initial state at $t=-\infty$ to the final state at $t=+\infty$.
$$|\psi(t=+\infty)\rangle = S |\psi(t=-\infty)\rangle$$
This matrix/operator is therefore a collection of complex numbers that are ready to calculate the probabilities of various scattering processes, e.g. those at particle accelerators if we talk about a realistic quantum field theory.
So the matrix acts on the Hilbert (sub)space of all collections of stable particles with arbitrary momenta – which may be assumed to be infinitely separated from each other so that their mutual interactions are negligible at $t=\pm \infty$ – and the result of the action belongs to the same space. In particular, the S-matrix of the Standard model may contain arbitrary combinations of electrons, photons, and other stable particles with arbitrary 3-momenta. However, it should also allow arbitrary protons, hydrogen atoms, or other stable composite particles! It's harder to compute the S-matrix elements involving composite states in the final state or the initial state but in principle, they're as good parts of the S-matrix as the S-matrix for electrons and photons.
(In fact, there was a whole S-matrix or Bootstrap program, initiated by Heisenberg and followed in the late 1960s by Chew and others – this program was a branch of research that inspired the birth of string theory. A basic philosophy of this program was that our calculations shouldn't be constructive; they shouldn't make a qualitative distinction between elementary and composite particles; all the allowed states and mass spectrum should be determined by some consistency conditions, "by itself", and that's where the "bootstrap" terminology comes from.)
One of the answers to your questions is therefore Yes, the full S-matrix knows about all the energy levels of the hydrogen atom and other things, too. In fact, the complete spectrum of stable particles in the QFT should be given before we try to determine the S-matrix elements in between them. Even unstable particles and unstable levels are "known" to the S-matrix because they manifest themselves as poles in the scattering of stable particles or objects.
Some quantum theories only have the S-matrix but they don't admit local "off-shell" Green's functions. Quantum gravity is a typical example. Off-shell Green's functions $G(x^\mu,y^\mu,z^\mu)$ etc. are not gauge-invariant because gauge invariance in gravity includes the reparameterization of the coordinates. So only the S-matrix for graviton (and other) states with particles obeying the on-shell condition $p_\mu p^\mu = m^2$ ($m^2=0$ for gravitons) are among the quantities that are manifestly Lorentz-covariant and well-defined in quantum gravity theories (such as string theory). So in the case of some theories, the S-matrix is the only calculable quantity among the conventional ones that makes sense. (Quantum gravity, i.e. string theory, isn't quite a [local] quantum field theory in the same spacetime; it's a generalization, and the fact that Green's functions aren't well-defined is an example of this fact.)
On the contrary, some quantum field theories (obeying all the locality etc. conditions of quantum field theories) don't have an S-matrix because they don't admit multiparticle states with widely separated and virtually non-interacting particles. All conformal field theories belong to this class. Conformal field theories don't have any preferred distance scale $L$, so if we ask what's the minimum distance between particles at which the interactions become negligible, the only answer may be $L=\infty$. It follows that the interactions in conformal field theories are always "long-range forces" – they act at arbitrarily long distances. In the case of weakly coupled theories with short-range (or Coulomb) forces, the possible scattering states didn't differ from the Fock space of all the excitations much but in CFTs, they do.
In CFTs, one can't define multiparticle states with isolated, nearly non-interacting, particles, and because the S-matrix is a matrix between such states and there are none in CFTs, there is no S-matrix in CFTs. (Or, less usefully but correctly when it comes to the literal rules, we could say that $|0\rangle$ is the only allowed asymptotic state for the S-matrix of a CFT and $1$ is the only scattering amplitude between two such vacua.) Instead, we usually want to calculate various correlation functions of local operators in CFTs. CFTs are always local; they can't suffer from the problem linked to quantum gravity in the previous paragraphs because a preferred scale similar to the Planck scale is needed for the locality to break down.
