Dear Peter,
I am grateful that you took the time to read my paper. It's probably not at all easy to make sense of. I sincerely want to figure out where you and I are making a different assumption. I just want to get to the bottom of the matter. Only thenn can we construct a correct theory.
Number 2 is easier, I'll start there. In a vacuum, photons don't accelerate. If we assume that black hole has nothing to eat, then this assumption is reasonable. Gravitational time dilation really refers to fixed points. I can't build a tower on an event horizon, but if we build a framework around the blackhole, then we can build a tower. The top of the tower is point A, the bottom is point B. Our laser is fixed to the top at point A, and the photons will fall to point B.
At your request, a train of photons is emitted from A and absorbed at point B. The invariance of the speed of light is true from point A to point B. As such, each photon will blueshift in accordance with the time dilation equation given in Appendix A.
You asked, "If we're 'at rest' near and wrt a black hole (a tricky assumption!) and see a string of photons being sucked in and accelerated, which one is accelerated first? Can I suggest the first/nearest one? So we will see the gaps between them open up as they get sucked in."
Now, we really can't want photons get sucked into a black hole. The best we can do is observe the very rapid redshift of photons trying to escape the black hole. When there are no more redshifted photons, then we can conclude that our laser has been devoured by the black hole.
Also, photons always travel at c, locally, even across signficant time dilation. Whether or not a black hole can violate the Invariance of c by making the photon's path "non-local from A to B", is a matter I would take up with experts. As for gaps between the photons, that would mean that time dilation has broken down and I would again defer to an expert on black holes. I can say that a breakdown in time dilation would also be a breakdown in conservation of energy. I'm not aware of any accepted black hole theory that permits a violation of conservation of energy.
Question 1:
The oncoming train emits photons that are blueshifted at the inertial frame of the observer. As the train passes, time dilation t'/t=gamma. As the train moves away, then redshift occurs. Scattering is not a form of transmission, it's really an obstruction to transmission. I think you mean that the photons are absorbed and re-emitted by scattering events. Even if there was a gas (fog) between the train and the observer, I'm just not seeing how time dilation can be violated.
I gotta get back to work. I'll think about it.
Once again, thank you for reading my paper.