What are the characteristics of early literature in diverse part of the world, including the ancient Near East, Greece and South Asia, China and Rome?

This is in some ways a tricky question, because it's asking for a discussion encompassing a broad geographical expanse, which risks erasing the particularities of different regions and traditions. It is a difficulty further complicated by the tendencies for academic and scholarly research to be focused along particular regions. One can expect a classicist to be able to comment in great detail concerning the particularities of Greek and Roman literature, but their knowledge might be far more fragmentary as far as it relates to the specificities of Ancient China or Southern Asia. For this reason, I will tailor my answer to the specific literary traditions one can observe in the Mediterranean World and the Near East.
First, and perhaps most importantly, I would note that storytelling seems to be a recurring theme in the human condition, one which far precedes the written form. There were stories long before any literature was ever written down. Ultimately, the oldest stories tend to have evolved and emerged out of a still older oral tradition and were later, eventually, written down.
Of course, oral stories need to be memorized, a feature which lends itself to poetry much more than to prose. Thus, we find most of the oldest great works of literature to be works of poetry (e.g., Homer's The Iliad and The Odyssey, The Epic of Gilgamesh, Beowulf). Poetry is an ancient and popular art form.
In addition, there are certain themes in ancient literature which tend to recur. In the Babylonian creation myth, for example, we find a succession of primordial deities, beginning with Apsu and Tiamat, with Apsu desiring to kill the younger generations, for which he is overthrown and killed. Later, Tiamat launches her own revolt and is defeated by Marduk, the patron god of Babylon, who is made master of creation as a result of his victory. (In retelling the essentials of the Babylonian creation story, I conferred with Barry B. Powell, Classical Myth: 4th Ed., Upper Saddle River, New Jersey: Pearson, 2004, 98–102). This narrative mirrors the more famous Greek creation story, as told in Hesiod's Theogony, with its own succession of gods, transitioning from Gaia and Uranus through to the Titans and then to the Olympians, with each generation overthrowing its predecessor. More broadly, in both cases, these are stories in which the forces of civilization overthrow these older forces of nature (see Powell, ch. 4). This theme, involving the conflict between civilization and nature (and the triumph of one over the other), can also be found in the exploits of Heracles, and perhaps also in the saga of Gilgamesh. There are also some more specific motifs that occasionally recur: the flood story, for example. We see it in the story of Gilgamesh, in the story of Noah, and in a much later version, there's the story of Baucis and Philemon, as told by the Roman Ovid.
As one final point of comparison, it's worth noting how many of the great heroes of the ancient Mediterranean and Near East are themselves tied to particular city-states and polities. Gilgamesh himself was the King of Uruk, while Marduk was the patron god of Babylon. When we look at the heroes of Greece, different city-states have particular heroes tied to them, such as Theseus, who was the king of Athens, or Oedipus, who was the king of Thebes.