When a photon passes through a transparent substance like glass, it is continually absorbed and re-emitted, retaining all its properties including entanglement.
If the polarization of two photons is entangled, redshift or blueshift does not change the entanglement of their polarization. You can even rotate the polarization of one of the photons, and they will still be entangled -- but in a different sense.
This really depends on details of the inelastic scattering. In the case of very simple inelastic scattering, the entanglement can be preserved, or it can be passed on to the atom. See the following broad overview of entanglement.
Entanglement really boils down to conservation of a conserved property like momentum, angular momentum, energy, and so on, in the context of quantum mechanics. For example, if a system of two particles has a net angular momentum of zero before the particles interact, then regardless of what happens in the interaction the angular momentum will still be zero after the interaction. So if one of the particles is measured to be spin up, you know the other particle will be spin down without ever having to measure it-- because otherwise angular momentum would not be conserved. In that case, where the "spooky" aspect of entanglement comes in is that, under the right circumstances, the spin of both particles is "indeterminate" until one of them is measured. Experiments have shown that, in fact, they are both "up" and both "down", until a measurement or other such interaction occurs. But conservation of angular momentum requires the possibilities to be "first one up, the other down" or "first one down, the other up". Conservation of angular momentum does not allow "first one up, the other also up" or "first one down, the other also down", because in those two cases the net angular momentum would not be zero. So it kind of appears that the two are in communication, though they really aren't. It's more accurate to say that their spin property is shared.