Is the classical Doopler Effect, for light shift, 1−v/c , exact? What is it an approximation of?

Chosing a reference frame in which the Earth is at rest and doesn't rotate
1) Does anybody know of such a publication?
2) I know that even such speeds of ${10}^{18}$ m/s are not in contradiction with relativity because a limiting velocity only exists for exchange of information, which apparantly does not occur.

Since $p^2=E^2-{\overrightarrow{p}}^2=m^2$ and
$E=h\nu =\frac{hc}{\lambda }$ and
$|\overrightarrow{p}|=\frac{h}{\lambda }$
we have that
$p^2=\frac{h^2{c}^2}{{\lambda }^2}-\frac{h^2}{{\lambda }^2}$
If I go to Planck-units ($c=1,h=1$), this becomes zero. Is this a correct thing to do?

From my understanding, we can do math in an accelerating frame of reference as long as "fictitious" force terms are correctly added. From this point of view, is there anything wrong with viewing the Earth as stationary, and the rest of the universe rotating around it, at least kinematically? And, if so, wouldn't several cosmological objects move faster than light in this frame of reference? How can this be?

If you have 2 flashlights, one facing North and one facing South, how fast are the photons (or lightbeams) from both flashlights moving away from one another?
Just adding speeds would yield 2C, but that's not possible as far as I know.
The reference frame here would be the place where the flashlights are and/or .The beams relative to one another.

If an observer were to rotate around a point at near light speeds, what sort of length contraction would he observe the universe undergo?

Imagine a space shuttle traveling through space at a constant velocity close to c. As the shuttle passes earth, a previously set-up camera starts broadcasting from earth to the shuttle. Since radio waves travel at the speed of light, the shuttle is receiving a constant transmission feed, assuming the camera is broadcasting 24/7.
Now, from what I have understood of special relativity so far, time will flow slower for the astronaut than for the earthlings. Hence, assuming $v=0.8c$, the astronaut will after 30 years have received a video transmission 50 years long!
Is my reasoning correct, that even though the transmission is live, the astronaut would actually be watching things that happened many years ago, while still receiving the "live" feed, which would be stored/buffered in the shuttles memory, thus making it possible for the astronaut to fast-forward the clip to see what happened more than 30 years after passing the earth?

Why is it hopeless to view differential geometry as the limit of a discrete geometry?
Classical mechanics can be understood as the limit of relativistic mechanics $R{M}_c$ for $c\to \mathrm{\infty }$.
Classical mechanics can be understood as the limit of quantum mechanics $Q{M}_h$ for $h\to 0$.
As a limit of which discrete geometry ${\mathrm{\Gamma }}_{\lambda }$ can classical mechanics be understood for $\lambda \to 0$?