How does gravity affect the wavelength of light? If, hypothetically, me and my rocket powered flashlight were falling straight toward the center of a black hole. The flashlight is a few kilometers behind me in our travels toward the center of the black hole, but since it is rocket powered, it manages to maintain the exact distance to me for a while.

The point is; The distance between me and my flashlight is constant as long as I am observing it.

The photons coming from the flashlight would obviously not be rocket powered - and they would be affected by the black holes gravitation.

Would the light I see from the flashlight be shifted towards red or blue, even though the distance between me and my dear flashlight is maintained?

If so; switching the positions of me and my flashlight, would it change the color I'd observe?

If we turn off the rocket on the flashlight, I assume it would be redshifted regardless of which were closer to the singularity, and the magnitude of redshift would appear to accelerate?
A partial answer: The lower the gravitational potential the slower the clocks, see Gravitational time dilation. This means, as long as your flashlight is behind you in constant distance, it appears blue-shifted to you. With roles exchanged (light below you in constant distance) it appears red-shifted.
By the heterogeneity of the gravitational field the shift grows with the time, while falling in constant distance.

This is called the Einstein shift, which is not the same as the doppler effect. The doppler effect is caused by an objects velocity relative to the observer.