PhD Student in Hebrew Bible and the ANE here. Some answers:
First, it needs to be far more widely known that literally nowhere in the Bible does it claim that the earth is 6,000 years old. Not once. This age is taken from adding the genealogies from the Bible together and assuming 1) that the time parameters being conveyed in the text align with our understanding of time today and 2) that the transmission history of the text itself has not led to alterations of any kind in the dates given.
In any case, onto the Mesopotamian evidence (which is what you're question is really concerning anyway):
AFAIK, there is no hard claim regarding the age of the world in the Sumerian, Babylonian, or Assyrian literary traditions.
Like we find in Genesis 1 (the earth is without light, night and day are undivided) there is a relatively consistent portrayal of the earth in its "before humanity" state in the Mesopotamian understanding. This is almost universally described in apophatic terms––meaning, it is described by what is lacking rather than what exists.
A couple of examples:
In the OB (Old Babylonian) poem Lugal-e, the conception of the pre-humanity world is on in which many major landmarks (the Zagros mountains, the Tigris and Euphrates as we know them now) do not yet exist. These are major features humans were experiencing on a day to day basis, and they convey the "before times" as a time during which these were not present.
Also from Lugal-e, we also find that irrigation agriculture, the primary means of supporting life at this stage in southern Mesopotamia does not yet exist. Following the defeat of the story's protagonist by the god Ninurta, agriculture is created. So (mythic) time is being conveyed as occurring before something that is integral to human subsistence has been invented.
In the Sumerian myth Enmerkar and the Lord of Aratta, the text begins by describing the construction of the Ur ziggurats as occurring "before Dilmun (Bahrain) had yet been settled." This is an important statement, as Dilmun is the occasional central local of creation in older Sumerian creation narratives. It's also the location where Utnapishtim (the character visited by Gilgameš at the end of the Gilgameš-epic) supposedly resides. Again, apophatic description of the world. Events are occurring before this really old, primordial place is settled.
So while there isn't necessarily an understanding of the exact age of the world ("it's old"), they did have a conception of the existence of the earth before humanity.
Sources:
Lugal-e (also known as Ninurta's Exploits or Ninurta and the Stones)
Enmerkar and the lord of Aratta
G. Rubio, "Time before Time: Primeval Narratives in Early Mesopotamian Literature," RAI 56