Notes on Sumerian religion:
I. The first creator: Nammu.
As far as I can tell, Nammu was considered the creatrix of the gods, even mother of the sky father An. A suggestion by the scholar Wiggerman was that her name meant creatrix based on what he called a "native" (ie Akkadian?) commentary. She was goddess of the deep waters of the Abzu, which etymologically means deep waters or deep knowledge (ab + zu). Her sign was Engur, which some say is a combination of the sign letters totality and secret, although one scholar tole me the pictographic letters lacked such a core meaning. I think she was portrayed as lizardlike or serpeantlike due to her association with the seas.
Question: Did all Sumerians always think Nammu was the uncreated creatrix of An? Did any of them suppose that someone else had made Nammu?
II. The supreme God An.
An's name etymologically means "high". It also could be used to refer to heavens. An's sign was Dingir, the sign for deity, made in the form of an 8 pointed star. Dingir meant not only "a deity", but heavens, the god An, or shining/brightness. A few scholars thought Dingir comes from Dimme, to create. An Akkadian linguistic commentary associated or equated An with Gira, the god of fire. So one might think Dingir was a reference to a heavenly fire (the root Tin + Gir) as a star, or something else to do with fire, as Dingir also refers to brightness. I think An was portrayed as humanlike.
Question: Could Dingir refer not only to An or mean "a god", but also refer to God Himself, like NTR, Tengri, and El can mean "god" or "God" in their own languages?
III. Monotheism.
This seems speculative in the case of Sumer, a problem being that we don't have texts on this topic from them. Babylonian theology in a way is a later version of Sumerian theology, and monotheism might be found in Babylonian religion, especially in its treatment of Marduk. But this aspect of the Babylonian mythology does not automatically mean that Sumerians thought this way.
IV. DNA and language
I guess they are pre-Semitic Iraqis. Their language roots have not been totally established. Maybe they had a mix of roots like Turkic and Dravidian. There are competing theories. I vaguely remember that 20% of Sumerian words are Akkadian and another 20% are Dravidian, and that Sumerians came from both north of Sumer and southeast of it.
V. Sumer in video games.
I think entertainment helps you feel the culture a bit. The best I've seen is Age of Empires. There is a Lost Sumeria campaign that looks great.
VI. Sumer in the movies
There does not seem to be alot for this.
I liked Heroes of the Bible: Tower of Babel,
Legacy - The Origins of Civilization - Episode 1: Iraq, the Cradle Of Civilization;
http://topdocumentaryfilms.com/legacy-the-origins-of-civilization; This movie was filmed in Iraq in 1991 before the first Gulf War. Good for its time.
(1991) - Filmed before Gulf War. Documentary
VII. Places to see.
Major cities have museums with Sumerian art and tablets.
VIII. Religions in the modern period
The Assyrian Church of the East and Chaldean Church were the two churches in Iraq for centuries. They can have some traces of Mesopotamian culture, but are principally Assyrian, although even the Assyrians were influenced by Sumer. In the Bible, Nammu seems to correspond to some deities in the seas like Leviathon or another one mentioned. She could most of all correspond with Tehom, the watery depths, who is a version of the Babylonian Tiamat. God's splitting the waters into heaven and earth can recall the splitting of the waters in Sumerian mythology.
The Yezidis are sometimes claimed to be a mix of Zoroastrianism and Mesopotamian religions, particularly Babylonian religion. A comparison has been made between the 7 angels who are manifestations of God for yezidis to a claim that suppoedly the Babylonians also taught 7emanations of God, but I couldn't confirm this. The story of the Garden of Eden, Noah's Ark and the Flood and the role of the snake is common to Babylonian, Sumerian, and Christian/Abrahamic religions, but this does not prove which religion the Yezidis took these stories from. For example, the appearance of these myths does not prove that they came to the Yezidis from Mesopotamian culture independently of Christianity or Islam. The idea of attending a Yezidi ritual with worship of Melek Taus / Shaytan is repellant to me.