Short answers, pulled from the story.
The sky appears blue because of Rayleigh scattering, confirmed by Albert Einstein in 1911. When sunlight passes through the atmosphere, oxygen and nitrogen molecules scatter the shorter blue wavelengths more widely than the longer red ones, sending more blue light to the human eye.
Blue eyes contain no blue pigment at all. Their colour results from the Tyndall effect, an optical scattering phenomenon in the stroma of the iris that reflects short-wavelength light outward, making the eye appear blue to an observer, much as Rayleigh scattering makes the sky appear blue.
Egyptian blue is the first known artificial pigment, produced as early as around 2500 BC. It was made by heating pulverised sand, copper, and natron together, and was used in tomb paintings and funerary objects in ancient Egypt.
Prussian blue was discovered in 1709 by German druggist and pigment maker Johann Jacob Diesbach. It arose from experiments involving heating dried blood with iron sulphides, and was initially called Berliner Blau. By 1710 the French painter Antoine Watteau was already using it.
Shuji Nakamura of Nichia Corporation, Isamu Akasaki, and Hiroshi Amano of Nagoya University shared the Nobel Prize in Physics in 2014 for the invention of an efficient blue LED. Nakamura had separately received the 2006 Millennium Technology Prize for his work on high-brightness blue LEDs, first demonstrated in 1993.
Ultramarine was produced by grinding lapis lazuli, a semi-precious stone mined primarily in Badakhshan province in northeast Afghanistan, into a fine powder. Its extreme rarity and the difficulty of extraction made it the most expensive pigment available. In Italian Renaissance art it was typically reserved for the robes of the Virgin Mary.