These colors come from the atomic fluorescence of particles high above the earth’s surface. As charged particles from the sun’s “solar wind” spiral through the earth’s magnetic field toward its poles, they collide with one another and with atoms in the earth’s upper atmosphere. The energy of such collisions can excite the atoms involved and cause them to emit light.
Actually, 96% of the light hitting the “other side of a water droplet” does pass out of the droplet. What you see in the rainbow is the 4% that reflects back from the far side of the water droplet. If all of the light reflected, the rainbow would be much brighter.
Not exactly. A puddle contains water, which reflects light directly. Light from the blue sky travels toward the puddle and illuminates it. As the light enters the water, with its higher refractive index, part of the light reflects. You see this light when you look at the surface of a puddle. But a mirage involves refraction (bending) of light. As light from the blue sky enters a regions of hot air near the surface, that light bends upward. You again see light from the sky, but bent upward by the air rather than being reflected upward by a surface of water. Since the two appear similar, you interpret the shimmering blue light of a mirage as coming from a pool of water. But it is just hot air.
The sun appears bluer when viewed from outside our atmosphere. The earth’s atmosphere scatters a substantial fraction of the violet and ultraviolet light in sunlight, leaving a reddened sun disk to our view. Without that scattering, the sun’s disk will appear to contain more blue and ultraviolet light.
The colors of flames can be deceiving because they involve emissions from particular atoms (which impart their own characteristic colors to the light they emit). However, a blue-hot object such as a star is hotter than a red-hot object such as a glowing coal in the fireplace.