Why is the Earth's shadow blue?
To my eyes, the belt of Venus looks purple, which didn't makes sense to me, as the very short wavelength of purple light should have been scattered long before arriving back there. Then all made sense when I realized that if you mix blue and red light you get purple.


You got that part right. Purple is not violet. Violet is a spectral color at the high frequency end of the visible portion of the spectrum, while purple is a non-spectral color on the line of purples that spans from nearly ultraviolet to nearly infrared. We see color as a wheel rather than as a line. One possible explanation is that the long wavelength ("red") cones have a secondary response in the violet end of the spectrum. Spectral violet light excites both the short wavelength and long wavelength cones, but so does a non-spectral mix of red and blue light.

And that's what you're seeing in the Belt of Venus -- a muted mix of red and blue, or a purple (or perhaps magenta, or a rosy pink).


  So the color comes from the blue mixed with the red light backscattered by the high atmosphere that is still illuminated by the Sun. However, I can't still figure out why the Earth's shadow looks blue.


The backscattered red light doesn't come from the high atmosphere. It comes from the troposphere. The high atmosphere (the stratosphere and above) contains very few particulates. The upper atmosphere is the only part of the atmosphere that remains sunlit above the eastern horizon after sunset. The sunlight hitting the upper atmosphere is Rayleigh scattered, so you see that very faint blue scattered light. You don't see the backscattered reddish light from the particulate-containing lower atmosphere at the eastern horizon because that part of the atmosphere is fully in the Earth's shadow above the eastern horizon.