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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).
Not directly. You have to square the velocity of a galaxy, then you must divide it by c (light speed). Then you must divide it by c once again. Then you have to subtract 1, change the sign, compute the square root.
Regardless of the direction of the galaxy in question.
You are very wrong here, I am sorry.
But that's beside the point. We want the right solution and that solution should be in a good agreement with all those Hubble pictures.