Yeah there is actually a few but not too many compared to the Greeks and Roman since the Mesopotamians mostly wrote down economic, trade, administrative, and property data on tablets (or at least that's what mostly survived from the ancient world). The most popular epic beside the Epic of Gilgamesh is probably the epic of Enuma-Elish, which is the creation myth of Mesopotamia. There are a few more here on this website but not too many epics survived on tablets from Babylon and the rest of the Mesopotamian city-states.

You may also be interested in the story of Atra Hasis (a flood account, which has attracted much attention due to its similarities with the Biblical one). The fragments of Debates are also quite fascinating (it has been suggested that some of them may also have been an inspiration for the book of Genesis). The “cycles” of Enmerkar, Lugalbanda and Dumuzid, fragmentary as they are, also contain fascinating bits. Oh, and also have a look at Ugarit — the time frame and even the culture are different, but their myths happen to be the best-known of the Ancient Middle East.

Quite a few! You may want to have a look at the Electronic Text Corpus of Sumerian Literature, which has a nice collection of Sumerian-language stories and poems including the Aratta poems(concerning the deeds of Enmerkar and his son Lugalbanda, kings of Uruk, and their relations with the land of Aratta), the "debate literature" mentioned by /u/GeorgiousFlorentinus, and various literary myths concerning Enlil, Enki, Inanna, and other gods. You may also find some of the hymns and sacred poetry interesting, especially the "city laments" such as the Lament for Sumer and Urim, as well as the "edubba texts", a collection of satires on education set in the Ur III period. Many of the types you might find in this literature will I suspect be quite familiar. Moving on to Akkadian literature, people have mentioned the Enuma Elish and Atra-Hasis, to which I would add for the sake of variety the various example of wisdom literature such as Ludlul Bel Nemqi and humorous texts like at the cleaners, about a know-it-all who who attempts to tell a cleaner how to wash his clothes. Last, I would be remiss if I did not mention cuneiform literature outside of Mesopotamia, including both the Ugaritic Baal cycle, other Ugaritic poetry, and Hittite myths like the Kumarbi cycle about sucessive attempts to overthrow the storm-god Teššub which is often compared to Hesiod's Theogony, the myth of Illuyanka, and the various "Vanishing God" myths. The best anthology for Akkadian literature is Before the Muses, and the Hittite myths are collected in Harry Hoffner's Hittite Myths