Why is an electron still an elementary particle after absorbing / emitting a photon? When an electron absorbs a photon, does the photon become electron "stuff" (energy); or, is it contained within the electron as a discrete "something"?
 A: When an electron absorbs a photon, it remains an electron and the photon disappears.  The electron energy and momentum are altered to account for the energy and momentum the photon was carrying.  For a free electron, it will not be possible to balance energy and momentum simultaneously.   There will have to be another interaction to make that work.  If the electron is part of an atom, it can transfer some of the momentum to the rest of the atom and it can balance.
A: Note that electron in isolation can never absorb or emit a photon.
It is only a system of 2 particles (*) than can.
P.S.
2 or more;
Theoretical consideration of electron in a static field requires something to create said field.
A: As Ross pointed out, two scenarios are possible: free electron / electron as part of an atom. They're treated in two totally different ways.


*

*Free electron: free electrons can't really "absorb" photons. They can collide with them, and some things can happen (this, for instance). Those types of collisions are described by QED and there are a bunch of conservation laws which govern them – lepton number, spin, energy, momentum, etc..

*Electron as part of an atom: this is probably the case you want. In its description the photon is not treated as a "particle" (in the QED sense, but even the classical "colliding spheres" analogy doesn't hold), and you sort of see it as a unit of energy (the "discrete something", a wave, pretty much) which gets delivered to the electron, which goes to a higher (read: less stable) energy level and then decides what to do: it can go back to where it was (spitting out another photon, i.e. energy), drift away (should the photon beat the electron's ionisation energy, this is the case for X and gamma radiation) or, ultimately, stay, until something else happens.

