Is it photons in -> (new) photons out? Or the same ones reoriented?
If we're talking about something like fluorescence, there's a fairy clear point where one photon disappears and another appears.
In a linear process like this, photons are not really absorbed by the material. In fact, the quantum behavior of photons is not relevant to the process, so you can just treat it purely as a wave phenomenon.
In cases like this, I would generally say that it is the "same" photon, but again, not really appropriate to think in terms of photons when there is nothing about the process that depends on quantization.
If I were to answer this question in terms of photons as small amounts of field oscillations, I would argue that these are "new" photons, due to the fact that the "old" ones induced oscillations in the dipole moment of the material, which then in turn radiated energy out as the "new" photons.
But you can just as easily think of it as the material "suggesting" a better direction to the field propagating through it, and thus reorienting it. This is just very difficult to imagine and describe, at least for me.
How light actually behaves is probably beyond the ability of human cognition (since so much happens in a billionth of a second)
This is not remotely true. The behavior of light is very well understood and relatively simple to model compared to other, less linear physical processes.
When you really dig into this stuff your realise stuff like the density of air is really an abstraction that doesn't quite fit what is actually going on.