Is there a doppler effect on the images of stars around rotating black holes? I'm an illustrator working on a project involving rotating black holes like those discussed in "Gravitational Lensing by Spinning Black Holes in
Astrophysics, and in the Movie Interstellar" by James, et al. 
I've found a good amount of information about the frequency and intensity shifts of the images of an accretion disk around a rotating black hole, but I'm having trouble figuring out whether or not the images of the stars around and behind the hole have similar shifts, or if those shifts only happen to light that originated within the immediate vicinity of the hole. 
 A: Stars orbiting black holes (I assume that's what you mean) and observed from afar will have their light doppler shifted due to (i) gravitational redshift; (ii) the relativistic doppler effect due to their orbital motion.
Effect (i) becomes more important the closer a star gets to the event horizon of the black hole. The redshifted frequency is given by
$$f_{obs} = f_{0} \left( 1 - \frac{2GM}{rc^2}\right)^{1/2},$$
where $r$ is the orbital radius and $M$ the black hole mass.
In practice, the maximum gravitational redshift will occur at the innermost stable orbit, which is at three times the Schwarzschild radius ($=6GM/c^2$) and amounts to a frequency redshift by a factor of 0.82
Effect (ii): In the relativistic doppler shift for a source moving at a speed $v$ at an angle $\theta$ (in the reference frame of the observer), the emitted and observed frequencies are related by
$$ f_{obs} = \frac{f_0}{\gamma\left[ 1 + (v/c)\cos\theta\right]},$$ 
where $\gamma = (1 - v^2/c^2)^{-1/2}$. This means that even when $\theta=90^{\circ}$ and the source orbiting the black hole is moving neither towards or away from a distant observer on Earth, there is a "transverse doppler redshift" of a factor $\gamma^{-1}$.
A star at the innermost circular orbit would have a speed of half the speed of light and $\gamma = 1.15$. Thus the redshift due to the transverse doppler effect would be a factor of 0.87 and almost the same as the gravitational redshift. At larger orbital radii, the gravitational redshift becomes more dominant.
On top of the net tranverse doppler redshift there will be a periodic modulation of the frequency as the source orbits the black hol. The amplitude of this will depend on the inclination of the orbit to the line of sight. At its largest, $\theta = 0$ and the redshift/blueshift will be factors of $\gamma^{-1}(1 \pm v/c)^{-1}$.
Thus for a source in orbit at the innermost stable circular orbit, the maximum frequency redshift would be a factor of 0.58, whilst the maximum blueshift would be a factor of 1.74.
NB All the above calculations assume a non-rotating black hole. The details are different for spinning black holes,
A: Yes, they absolutely would.  In general, a light-ray which passes at a minimum distance $x$ to the BH will have all of the same effects as a light-ray emitted at the same distance $x$.
A: In addition to what has been being discussed here, we should also make a clear distinction between the Relativistic Doppler Shift and the Relativistic Beaming; Doppler Shift is due to the change in the frequency of the emitted photons whereas the Beaming is due to the change in the intensity of photons regardless of their shift in the frequency. I have put together a note where I have contrasted these two effects where GR effects were not included: https://www.ashlarstem.com/ashlar-blog/relativistic-doppler-shift-vs-relativistic-beaming
Interestingly, we see here that high speed sources moving towards an observer may create photons beamed up towards the observer but red-shifted. How is that possible? It is all in above link.
