The galaxy (the Milky Way) moves around the universe at about 5% of the speed of light. Why do I see the same walls? Where is the Doppler shift? One wall should be slightly red, and the other - slightly purple.

People who have at least received school education look at you with pity, young man.
With the help of simple visual observation, it is easy to notice that RELATED to you, the walls are not moving at this speed.