"One of the themes that we have returned to is the question of whether some behaviours or traits of humans are innate, or whether they're produced by culture and/or environment. With reference to the ancient cultures we have studied, to what extent are humans hardwired for certain behaviours or traits, and to what extent are we products of our culture and/or environment?"

This is a very broad question. The good news is that there are a plethora of examples from which you can draw a response. The bad news is that, with some human traditions, it is difficult to determine exactly when traditions became a part of collective culture and how they developed.
Let's consider, for example, the traditions of funerals and burial. The historians Will and Ariel Durant wrote, in the first volume of The Story of Civilization, Our Oriental Heritage, that many early cultures engaged in cannibalism and, therefore, had no need to bury their dead. The Durants mentioned how human flesh was a delicacy among "the Irish, the Iberians, the Picts, and the eleventh-century Danes" as well as among people in the Upper Congo, where "living men, women, and children were bought and sold frankly as articles of food." Funerals were unknown among these people, who had no shame about drinking blood or expressing a preference for human flesh over that of other animals. Among these people, funerals seemed like "an unnecessary extravagance."
As the concept of the soul developed, however, as an entity that exists long after the flesh has wilted, cannibalism came to be seen as immoral. The Sumerians were among the first to develop burial mounds, but these were no more than crude graves. The Egyptians are the first known civilization to have developed elaborate funerary rites. The pyramids, after all, are tombs. Pharaohs believed that the soul, or ka, did "not die with the breath" but "would survive all the more completely if the flesh were preserved against hunger, violence, and decay. Thus, the body was not only entombed and placed within a pyramid, but it was also mummified to keep it as close as possible to its original condition. This process of preservation set a precedent for our present-day practice of embalming.
It is difficult to argue if humans are "hard-wired" not to engage in cannibalism; for, at one time, this was a common practice. Also, there are a few instances in which we discover that cannibals exist, even in industrialized nations. However, there is a common consensus across cultures that human life is more sacred than that of other animals, due to the higher intelligence of human beings and the belief, among many people, that humans have a soul.
Funerary rites have evolved both from our belief in the human soul, as well as from our wish to acknowledge the personal impact that people have had in our lives, particularly the memories we have created with them. Some people extend this consideration to pets as well, which has led to the creation of pet cemeteries and funerals for cats and dogs. Thus, it seems that our understanding of the soul and of the value of other living beings is constantly evolving, more as a result of advances in cultures than of any innate understanding about human life.