How well can we calculate atomic spectra from first principles? Calculating atomic spectra means to me that one takes a Hamiltonian neglecting some effects known to be present (any such Hamiltonian will always neglect something, be it relativistic effects or even quantum gravity) and tries to approximate its (math.) spectrum. 
I've only every seen this scheme applied to Hydroge: The most naive, exactly solvable Hamiltonian gives rise to the Bohr formula of energy levels. One can then add to this "relativistic corrections to the dispersion relation" while keeping the theory non-relativistic, as well as couplings between the electron spin (or rather magnetic moment) and angular momentum. These are referred to as Fine Structure and provide corrections that are smaller than the Bohr energies by a factor of roughly $\alpha^2$ (Fine Structure Constant). There is then the Lamb shift, smaller by another factor of $\alpha$, that in a sense incorporates the fact, that the em-field is quantized without actually quantizing it, and the Hyper-Fine Structure from the magnetic interaction between dipole moments of electron and proton.
Alternatively, one can consider the Dirac equation (and Dirac Hamiltonian), which is fully relativistic and should be phenomenologically consistent in the regime where no charged particles are created, despite being not bounded below. There is also the Fierz-Pauli model, which considers non-relativistic particles in external potentials inside a fully quantized, non-self interacting electromagnetic field. I'm not sure what the respective predictions are (I think it's hard to work out the spectra of these models) and how to evaluate the precisions of each of these descriptions. If anyone knows sources that compare all of these calculations to good experimental data, I would be very interested even for Hydrogen.
In 1951 Kato proved that the Coulomb Hamiltonian for an arbitrary (finite) number of moving positive and negative charges, with instantanous Coulomb potentials acting between them, is self-adjoint. In principle the same calculations should be applicable to other atoms, even if it may be harder to work out the spectra of those Hamiltonians, perhaps by some approximations or numerical methods. I imagine that similar schemes to the ones above should be possible to apply to more complicated atoms than Hydrogen, without too much conceptual difficulty. My question is: 

Which of these models have been applied to different atoms and how good is the agreement with experiment? What effects are more important in larger atoms, that may be neglected for Hydrogen?

 A: The state-of-the-art theory for the description of atomic and molecular physics is quantum electrodynamics or QED. However, for light and few-electron systems, where the velocity of the electrons is relatively low, often a simplified form of QED is used, namely N-QED ("N" standing for "Non-relativistic"). The idea here is to treat the interactions arising from relativistic and QED theory perturbatively and evaluate their terms using the solutions of the nonrelavistic Schrödinger Equation. Under these assumptions, one can write the total energy as an expansion in the fine-structure constant $\alpha$ which is a measure for the (classical) electron speed
$$
E(\alpha)=\mathcal{E}^{(0)}+\alpha^2\mathcal{E}^{(2)}+\alpha^3\mathcal{E}^{(3)}+\alpha^4\mathcal{E}^{(0)}+\ldots
$$
where the term $\mathcal{E}^{(0)}$ represents the complete non-relativistic energy, the term proportional to $\alpha^2$ represents the relativistic corrections, the term proportional to $\alpha^3$ represents the dominant QED interactions and higher terms in $\alpha$ represent higher-order QED corrections. Note that some authors have an additional factor of $\alpha^2$ in the equation above because each energy term itself is proportional to the Rydberg constant which scales as $\alpha^2$. For atoms, people like Gordon Drake and Krzysztof Pachucki have been able to calculate energy levels of light atoms using the values of the fundamental constants and the number of protons, neutrons and electrons as only input for their calculations. In case of one and two electron atoms, the theory is so advanced that the uncertainty of the calculation is dominated by the (experimental) uncertainty in the values of the fundamental constants such as the Rydberg constant or the proton-to-electron mass ratio. The simpler an atom is, the more terms of the expansion can be evaluated and the more accurate the energy can be determined. Heavier atoms are more difficult, but even in the case of the boron atom (with 5 electrons) theory was able reproduce experiment with nearly parts per million precision. 
When the nuclear charge and number of electrons increases, the assumptions made above no longer hold and the expansion of the energy in terms of the fine-structure constant does not longer converge. Similar arguments are used for diatomic molecules such as H${_2}^+$ and H$_2$. Especially for the one-electron molecular hydrogen ion, theory has an accuracy of 2 kHz or $8\times 10^{-10}$ kJ/mol.
