It is not my conviction that the creation account in the first chapters of Genesis is intended to provide literally correct and scientifically adequate descriptions of the world's coming into existence. Other creation myths in the ancient near eastern context don't have this intention, and in light of present scientific knowledge it is not plausible to suppose Genesis does as well. I think rather the goal of the creation story was to offer a theological account of the nature of reality, and of the world's (and humanity's) relation to God, in contradistinction from the creation mythologies of other civilizations surrounding the ancient Israelites.
For instance, one manner in which the Genesis creation myth distinguishes itself from, say, the account of the creation of the world in the Babylonian Atrahasis myth is the nature of divine-human relations. In the Atrahasis, humanity is created because of a labor dispute among the gods. The Igigi, a sort of worker class of deities, grow tired of the menial labor to which they have been assigned, and they begin to disturb the rest of the divine bourgeoisie with their complaining. In response, a scapegoat is slain and from its various body parts and blood the human race is made. But humanity is made for the purpose of working the labor of which the Igigi had grown tired. On this view of things, humanity's condition is fundamentally a miserable one: relative to the gods they are slaves, this is their proper place, and there is no point complaining about it; humanity exists as a means for the gods' own ends, and they are not valued in their own right.
Genesis paints things differently in the extreme. Here God does not create out of any need, but only out of the pure power of his word to bring things into being. He utters in Latin (because surely God speaks Latin): Fiat lux! and est lux. And it is clear that he creates the animals and humanity for their own sake, not for his, since he gives them the entire world to fill with their children. He commands them to prosper and to flourish and to multiply themselves, so that they too may take part in the wonder of creation. When he sees that Adam is alone, he creates an appropriate helper for him, concerned for the good of the man (Gen 2.18). He provides for humanity and the animal kind a naturally replenishing source of sustenance.
In every way, then, Genesis and Atrahasis differ on their conception of the relations between God and his creatures. In the former, God creates his creatures for their own sake, and his fundamental disposition towards them is a kind of committed benevolence with no self-interest. In the latter, the creation is made so that the life of the gods is made easier, and there is no fundamental commitment on the part of all divinity to the creature. Genesis speaks against this view, as well as again Christian variants of the Atrahasis theology, such as an Augustinianism according to which God creates some persons purely for the sake of displaying his justice in punishing them eternally for their sin. No -- God is fundamentally committed to the good of the creature, and this without any self-interest.
Another difference is the manner of creation. As I've said, in Atrahasis the creation of man comes about through the ritual murder of a divinity; there must be a scapegoat for there to be humanity. But in Genesis God creates of his own power, subduing a kind of cosmic watery chaos, and imposes order upon everything. This is ultimately the point of the seven-day sequence: not that creation happened in seven literal days, but that everything has its proper order and place in God's world. Waters are separated by sky and land, animals located in their various places, and humanity created last, male and female, because there are natural proper places for everything. Mixing in various ways goes against God's intentions -- at least mixing where God did not command it (cf. Gen 2.24).
Genesis likewise offers a kind of animal ethic. The fact that the diets of animals and humans were vegetarian suggests that violence against other animals is not a part of God's intentions for the creation. Sacrificial practices postlapsum notwithstanding, the Genesis story puts forth God's ideal of peace among his creatures. The creation does not occur through violence because violence is not God's intention, it is not permanent, and it is not good -- in whatever form it takes.