How do scientists know the age of the CMB? And how does it correlate to the age of universe?
By combining what is presently well-understood about the interaction between particles and radiation at extremely high temperatures with what is also known about the thermal history of the universe, mathematical analysis of the characteristics of the CMB allows astrophysicists to accurately estimate both the age of the universe and the age of the CMB itself.
In this sense, the characteristics of the CMB tell us the age of the universe.
An excellent exposition of this topic is contained in Stephen Weinberg's book, The First Three Minutes, and in a variety of more current papers in the literature specifically about the CMB.
The motion of stars galaxies and clusters of galaxies is measured by observing changes in the spectrum of light arriving on earth of specific atoms existing on the celestial bodies which emit light.