The first has to due with scattered light. The nitrogen and oxygen molecules in the atmosphere act like little dipoles, scattering some of the light passing through. The amount of light scattered increases as the fourth power of frequency (or inverse fourth power of wavelength). The process is called Rayleigh scattering (yep, same guy as the last rule).
Because of that strong wavelength dependence, our blue-sensitive cones receive 3 times as much light scattered by nitrogen and oxygen as our red cones, and when we look up at a cloudless sky in the day, we see the sky is blue. Here are the response curves for the three types of cones and the rods in human vision. (That's right, you actually have four color vision, but you can only see blue-green (498nm) with your rods when it's too dim for your cones to see.)
But now imagine what a hawk sees. It has another color channel at 360nm, which sees 6 times as much scattered light as red. When looking up, the sky will appear more UV than blue. But there is more to it than that.
Rayleigh scattering is not isotropic. The dipoles scatter most strongly at right angles to the incoming light. When you look up at the sky, the intensity of blue changes from near the sun to 90 degrees away. It's a little hard to see because when looking up you also see Mie scattering which adds yellow light to the visible sky near the sun. But when you look down, like a hawk does, Mie scattering isn't an issue (instead you have less air Rayleigh scattering and more ground signal). The overall color gradient from Rayleigh scattering that a hawk sees looking down will be twice as strong as the color gradient we see, because the hawk sees in UV. In direct sunlight, the hawk has a measure of the sun's position whenever it is looking at the ground, even when there are no shadows to read.
The other consequence is axial chromatic aberration. Most materials have dispersion, that is, they present a different index of refraction to different wavelengths of light. One consequence of dispersion is that blue focusses in front of red (for lenses like the eye). I used to think this was a bad thing. But if your cones tend to absorb light of one color and pass light of the others, and your resolution is limited by the density of cones, a little chromatic aberration is a good thing, because it allows you to stack the UV cones in front of the blue cones in front of the green and red cones, and they all get light focussed from the same range. I know that retinas have layered stacks of rods, cones, and ganglion cells, but I don't know that any animals, hawks in particular, have actually taken advantage of axial chromatic aberration to stack cones of different colors. It's certainly something I'll be looking for now.