What is the origin of the custom of Thay wizards shaving their heads and why did they choose the red as their color?
The answer regarding shaved heads is simple, that they originated among the Mulani people, who shave their heads.
Many traditions of the old Red Wizards were lost in the transformation and modern Red Wizards don't maintain the same Mulani traditions of shaved heads and tattoos that the old organization did.
Mulan often had little body hair, ranging in color from black to dark brown, and all nobles and many other Mulan of both genders shaved any hair they did have.
I have not found any reference yet to why they picked "Red" as their color. Historically, the reasons for a given organization to pick a particular color have ranged from the symbolic to a matter of what dye they had on-hand. Red is typically seen as a passionate, even violent, color. It's the color of blood, of arousal, of rusting iron, all themes that could have appealed to a group planning to establish a magocracy that prides itself in being able to do what it wants due to its power. Or maybe they just all happened to be wearing red when they broke off and it became a tradition.