What happens after a photon is absorbed by an electron Say you have some material with enough bandgap for a photon to be absorbed by an electron. The electron goes to a higher energy state. What happens after that?
I would expect the electron to go back to a lower energy state and release a photon. Does it release only one photon? Does it have the same energy, i.e. can it release two photons whose energies add up to the original photon? Does it scatter? In the classical case we would expect to have reflection/refraction. What happens in the quantum case.
Also, now that there is a free electron state available, can a second electron fall to it and release a low energy photon etc.? Or does the highest energy electron try to fill that state?
 A: One has to keep clearly in mind that a bound state, like an atom or a molecule or a crystal lattice, involves more than electrons. There are protons in the nuclei of atoms generating the potential well which we usually assume that the electron is trapped in. It is a convention, due to the fact that the mass of the electron (~0.5 Mev) is so much smaller than the mass of the proton ( ~0.938 MeV ) that even for the simple hydrogen atom, it is usual to talk of electron energy levels. The large mass of the nucleus makes the assumption reasonable, that the center of mass of the problem is on the positive charges and it is the electrons that are moving in orbitals. The energy levels characterize the atom as a whole , they are the fingerprint of the atom, not the electrons or the nuclei individually. 
With this in mind

I would expect the electron to go back to a lower energy state and release a photon. 

That is what happens, the excited atom deexcites with the emission of a photon.

Does it release only one photon? 

If there exists empty energy levels below the excited energy level, then the energy could be released in a cascade of photons.

Does it have the same energy, i.e. can it release two photons whose energies add up to the original photon? 

These will add up to the energy of the original photon .

Does it scatter? In the classical case we would expect to have reflection/refraction. What happens in the quantum case.

No, it does not scatter within the time constants of the interaction. Once a photon is emitted, then of course it can scatter on other atoms.

Also, now that there is a free electron state available, can a second electron fall to it and release a low energy photon etc.?

There are many "free" energy levels over the neutral filled levels for all atoms.
An electron may be caught in an energy level above the neutral atom's level, creating a negative ion, depending on the filled energy levels of the atom.

Or does the highest energy electron try to fill that state?

No, it is energy levels one is talking in quantum mechanics, of the bound state. The electrons and the nucleus are not independent agents.
