Your concepts of optics, the Doppler effect, and light waves does not seem to have a problem here. However, there is an issue with your concept of the universe. The universe is expanding, but it does not have a "center" that it is expanding away from. The problem probably stems from a misunderstanding about the Big Bang. Popular culture has misdirected many people to think the Big Bang was a point; the origin of some explosion. However, it was actually a moment in time (the first moment in time) when the proper distance between all objects was zero. A common analogy is an expanding balloon. Consider the surface of an inflating balloon. On the surface, you cannot say that there is a center away from which everything is expanding. In this way, the universe is like the balloon and we are as ants on its surface.
Having said all of this, the expansion of the universe does affect the frequency of light that we see. The universe is expanding such that objects further away recede from us faster. This causes light that comes from further away to be more redshifted than from local sources. See Cosmological Redshift. This shifting of the frequency is not, however, frame dependent. To every inertial frame of reference, it would appear that space is expanding away from the origin the same way and that light is redshifted the same amount with distance.