How do "civilizations" develop and to what degree was the code of Hammurabi fair and just?

The first question is extremely broad. On that very general level, one can say that some civilizations subsist by hunting and gathering. These migratory groups tend to stay in small bands rather than forming large groups because hunting and gathering can only support low population densities. Due to their migratory nature, they develop little in the way of technology. Other peoples undergo the "neolithic transition", learning how to domesticate plants and animals. This more efficient food production is the precondition for urbanization. Agriculture produces a food surplus, meaning that classes of artisans develop who improve technology. Large urbanized cultures require laws and complex political organization.
The Code of Hammurabi strove towards justice as it would have been understood in ancient Mesopotamia. It emphasized the duty of the king to protect the weak, the poor, and widows and orphans, and to act fairly. It created fixed punishments for various sorts of crimes. As is typical of its period, it treated people differently depending on their gender, whether they were slaves or free, and whether they were commoners or nobles. However, despite this, its emphasis that the wealthy and powerful should not be allowed to act with impunity is evidence that the code was striving towards a sense that at least within a given social class all people were equal before the law.