What is the Ancient Near Eastern cosmology of ancient cultures in general historically, apart from Genesis? In other words, what did ancient...

This is a
fascinating question. From what I've read (I am drawing from the analysis done by Barry B.
Powell, as found in Powell, Classical Myth, Upper Saddle River: Pearson
Prentice Hall, 2004), it does appear as if a fair number of these Near Eastern creation stories
tend to share a lot of common ground with Hesiod's Theogony (the famous
Greek creation story). They seem to reflect many of those same themes of cosmic warfare, by
which divine rule changes hands through way of conquest.

In the Hittite
creation story, for example, as it is told in Kingship of Heaven, there is
an entire succession of gods, who give way to one another by way of conquest: Alalush loses
power to his servant, Anush, who in turn loses power to Kumarbi, who will himself, in turn, be
overcome by his son. (A far more detailed account on these Hittite creation stories can be found
in Powell, pp. 102-104)

Meanwhile, there is also the Babylonian creation
story, which also has striking comparisons to Theogony. Originally, there
are the primordials Apsu and Tiamat, the original progenitors from whom the later gods are born.
However, after Apsu determines to kill his children for being noisy, he is put under an
enchantment by one of his would-be victims, Ea, and killed. Ea will later father Marduk. Later,
Tiamat herself will turn against her children, gathering monsters to aid her. Overwhelmed at her
advance, the gods are forced to turn to Marduk, offering to give him supremacy if he would
defeat Tiamat. In so doing, he emerges as ruler of the world. (The Babylonian creation myth is
recounted in Powell, 98-102)

As Powell writes of the Mesopotamian tradition,
drawing parallels between it and the Greek creation story:

Mesopotamian and Greek myths alike report a cosmic history that begins with mighty
powers of nature and ends in the organization of the universe as a monarchic, patriarchal state.
Both mythical traditions make use of the motifs of succession and dragon combat. (Powell,
106)

He likewise sketches out powerful similarities
between the Hittite stories and the Greek creation myth (106-107). In each of these traditions
(Hittite, Babylonian, and Greek), there is a very different understanding of creation and the
cosmos than what can be observed in Genesis.