Praying Mantis are probably the most intelligent of all insects, as far as insect intelligence goes that is.
They have binocular vision so they can judge the distance to their prey before striking.
The three tiny simple eyes they have between the two large compound eyes are for light detection only and can't actually "see".
A Praying Mantis moves and cocks its head at you because it's watching you move around in front of it and it's keeping track of you.
The small black dot you see is, I believe, only a diffraction of light and dark that you see inside the eye which "appears" to be following you as you move. This illusion is because the dot is a spot that's always toward you as you move. It is not following you, it just "appears" to be doing so because of light diffraction within the many cells of the compound eyes.
The color change in the eyes is apparently due to the time of day. Dark at night and green during the day.
Praying Mantises are extremely beneficial and help control the insect populations of the world.
Professional Wildlife Cinematographer, Photographer, & Naturalist 36 years.