It's not really the air that is doing the discoloring. The discoloring is already there and it's from the oils in the skin and fat of the animal.
When you soak the skull in water long enough, a certian amount of the oils are leached out of it making it appear as though it's not as discolored.
Boiling them with Sodium Carbonate pulls out even more of the oils, and what you don't get out basically stays and the peroxide discolors it even more by turning it white.
Kinda like when us guys think were cleaning the toilet. Pour some bleach in and in an hour or two flush it. It's nice and white and looks clean.
The crud is still there......you just bleached it white.
Peroxide and or bleach will kill a certian amount of germs which in turn eliminates any odor that might be left, but you should still seal them by painting or clear coating to preserve them the way they are and prevent any oil left from resurfacing and bringing any odors with it. Which will eventually draw insects.