Does the Earth's rotation affect redshift measurements? How do you account for the spin of the earth in the mechanical sense, when measuring redshift?
Does the relatively slow speed of the earth have a major change in the measurement because of distance?
 A: The average orbital speed of the Earth is 29.78 km/s (with the rotation speed being a hundred times smaller), so that clocks in at about 1 part in 10,000 with respect to the speed of light.
When redshift is used to measure astronomical distances, the redshifts are often of the order of a few percent or more, which is a great deal higher than the precision limit $\delta z = 10^{-4}$ imposed by the orbital speed of the Earth on redshift measurements that do not account for it.
If and when astronomers measure redshifts closer to that limit, then the orbital speed of the Earth is obviously taken into account, and that precision limit is removed.
A: It does have an effect, but whether or not you have to worry about it depends on how sensitive your measurements are. The rotational period of the Earth is about $24\operatorname{hours}$. Depending on your latitude, that could contribute to your velocity as little as $0\operatorname{m}\operatorname{s}^{-1}$ (at the poles) or as much as $464\operatorname{m}\operatorname{s}^{-1}$. Worse, the effect that this has on your observation will depend on the angle between the direction to the source when the observation is made and East along the ground. So, it's a little complicated.
That said, $464\operatorname{m}\operatorname{s}^{-1}$ corresponds to a Doppler shift between $\pm 1.55\times 10^{-6}$, so only the most sensitive measurements will be affected, and mainly if the source is near the East/West horizons.
A much more important Doppler shift to account for in most cases is the one from the orbital revolution of the Earth about the Sun. That has a speed of about $29.8\operatorname{km}\operatorname{s}^{-1}$ and depends on the angle between Earth's motion and the source (worked out from the source's position in Ecliptic coordinates and solar elongation). That effect is up to $\pm9.9\times 10^{-5}$, so even then the measurement will have to be pretty precise for it to matter.
Another redshift that I've never seen discussed in this context (because it's sooooo tiny) is gravitational redshift. Basically, when a photon climbs out of a gravity well it gets redshifted, and when it falls down one it gets blueshifted. The effect is so small that I've only seen it used in the context of whole galaxy clusters affecting cosmic microwave background photons (see: Sachs-Wolfe effect).
A: The rotation speed at the Earth is about 0.46 km/s at the equator. The spectrum of a celestial object will be blue- or red-shifted by this amount if observed close to the horizon from an equatorial observatory. For other configurations, the effect will be smaller, depending on the altitude and azimuth of the source and the latitude of the observatory.
This is a trivially small correction to cosmological redshifts, which are of order thousands or millions of km/s, however it is an importantly large correction that needs to be made to doppler measurements of velocities of stars in our own or nearby galaxies. Indeed the size of this effect is much larger than the typical radial velocity signature due to orbiting exoplanets around a star (typically of size 1-20 m/s) and so it must be corrected for in these circumstances.
In practice, because it is easy to calculate, the correction can be routinely applied to all measured redshifts, in addition for correcting for the Earth's motion around the solar system barycentre, though it is often not done in cosmological surveys because the uncertainties in the redshift measurements are usually much larger than either of these effects.
