Redshift, Earth's orbital speed, and the speed of light

[iantresman]

The Earth is moving around the Sun at around 30km/s, a change of 60km/s over the course of the year.

1. Presumably, measurements of the speed of light would show a redshift up to ±30km/s? Are redshift measurements this accurate?
2. Likewise, the speed of the Solar System around the Milky Way, is estimated to be about 230 km/s, and the Milky Way is moving at some speed through the universe. Do these need to be these factored into redshift measurements?

 

[mfb]

Presumably, measurements of the speed of light would show a redshift up to ±30km/s? Are redshift measurements this accurate?

It depends on the measurement. The best spectrometers are sensitive to about 1m/s for single stars, those are used to search exoplanets.
The orbit of earth, the rotation of Earth (still several hundred meters per second) and the influence of the moon (~10m/s) are well-known, so they are always taken into account where they are relevant.

Likewise, the speed of the Solar System around the Milky Way, is estimated to be about 230 km/s, and the Milky Way is moving at some speed through the universe.

There is no "speed through the universe" as all motion is relative. We have some known velocity relative to the local cosmic microwave background, this is taken into account.