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Yes. You were my inspiration for this problem.
How exactly is everything baked into z?
The mass is increased and volume is decreased by factor gamma.
So the density is increased by gamma squared.
Do we really see that?
You can use a light wave as a clock. The ratio of frequency that the light wave is emitted at to the frequency we perceive is 1+z. Thus, the ratio of the time we observe a galaxy for to the amount of time that elapsed in the galaxy's proper time is also 1+z.
For non-relativistic motion, the speed is approximately proportional to the redshift, but as speeds get higher, that breaks down. Apparent slowdown in terms of v will involve a Lorentz factor, but in terms of z it will not, because of the definition of z being in terms of apparent slowdown (of light).