Why does the gas from dry ice flow down?
The gas flows downwards because it is more dense. It is more dense both because $\ce{CO2}$ is heavier than $\ce{N2}$ and $\ce{O2}$ and because the gas is colder than the ambient air.

The mist is water in the surrounding air condensing into microscopic droplets due to the lowered temperature.  The amount of mist depends on the humidity of the air; in perfectly dry air there will be no mist.

All of the gases involved are colourless. The mist appears white because it is in small particles and so it scatters any light falling on it.  It is the same colour as "natural" mist and for the same reason.