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Could also be redshift too. I don't know enough about it to know if we'd notice it over such a short distance and of a constant source, though. Definitely noticeable as reciprocity failure during long exposures in photography.
No, that's not at all what redshift is.
And neither redshift nor dopplershift would have that much effect on light at the speeds we're talking about.
Besides the sun's color on earth it's not a shift of wavelengths, it's a subtraction of wavelengths, as you easily can see in a spectrogram.
Do we experience Doppler shift with our own sun? I would assume as we are in a stable orbit we wouldnt
Strictly speaking since our orbit isn't a perfect circle we do move towards and away from the sun, so there will be some level of redshift. At those speeds there's really no chance of seeing it without specialised tools, in the same way you don't see redshift from a car driving past you.