What exactly happens in the basic QED Feynman diagram? When a photon is absorbed by an electron, I think that the following things happen:


*

*The electron changes in momentum, angular momentum and energy.

*The phase of the electron wave function changes by a fixed angle given by the coupling constant.


Is this correct? Is this complete? 
 A: A Feynman diagram is not depicting a physical process. Virtual particles do not exist. Strictly speaking that particle is not 'a' photon. It's just there to signify that the interaction is mediated by the electromagnetic gauge field $A^{\mu}$.
Physically, what happens is that the electron is always travelling in an ares where it feels the influence of $A^{\mu}$, because the latter has infinite range. So it is not a free particle, but an eigenstate of the Hamiltonian including both free terms and  $A^{\mu}$. Its state will therefore evolve into a different momentum state, and will acquire a phase shift, as if it scattered a single photon.
To calculate the maths of this, you would have to resort to perturbation theory, which is where all the confusion originates from. 
Perturbation theory needs you to choose a set of 'unperturbed' states, usually taken as the free electrons i.e. solutions to the Dirac equation. The $A^{\mu}$ is then treated as a perturbation, and energy and wavefunction shifts can be computed to higher and higher level of accuracy with higher order perturbative terms. Each term in this series can be pictorially represented by a Feynman diagram, which is what makes them useful.
A: As for the first point, any of the quantities by which the electron is characterized are changed due to conservation laws, which are the 4-momentum conservation law, angular momentum conservation law, charge conservation law. The photon doesn't carry electric charge but it definitely carries the 4-momentum and the angular momentum. 
As for the second point, you should note that there are no such meaning as the electron's wave function during the scattering. Within the Feynman perturbation theory, instead of the wave functions we're dealing with the Fock multi-particles states, and Feynman diagram just tells you how to write the amplitude corresponding to transition from two different Fock states. Therefore it isn't completely correct to talk about the wave function's phase. You can, however, talk about it when considering the scattering of the particle in the external field, which is not so strong that particles creation occurs.
Let me also clarify the SuperCiocia statement about the Feynman diagrams. In order to talk about the physical intermediate states during the scatterings (including also your example) you should use the so-called old-fashioned perturbation theory (you can find information about it by googling), in which the time (and therefore the energy) are initially "dedicated". The sum of all of the possible intermediate states is possible to be rewritten in Lorentz covariant form, which is known as Feynman perturbation theory. In this form the time is of course not "dedicated", and therefore there is no meaning of intermediate states. We introduce the propagator instead, and it corresponds to something like the particle travelling forward in time "plus" particle travelling backward in time.
