Is there a way to access object properties or any other properties from the material node editor? Can a value assigned in Material Node editor be changed using python while the game is running?
The easiest solution is to use Object Color. This can be accessed both by python and by the node editor, and gives you access to three per-object values.
In normal BGE you have to create a shadeless white material with object color enabled. Remember to make the diffuse color white, otherwise the color is multiplied (by default) by 0.8!
Slightly unclear from the image is that there are two materials: the shadeless object color one, and the node one.