1. 'Mesopotamia' as a geographic term has never gone away; regarding 'Ancient Mesopotamia' as in the cultures that existed there until the coming of Alexander (as good an endpoint as any, though there's no clean break), it depends. Knowledge of the Assyrians, Babylonians, and 'Chaldeans' survived in ancient Greek texts and in the Bible. The Sumerians and Akkadians were rediscovered in the second half of the 19th century, after British, French, German and American archaeologists began digging up their cities; they were given those names when Assyriologists came across the terms Agade and Shumeru in cuneiform texts. Paul Collins's The Sumerians is mostly about how our understanding of the Sumerians grew to what it is now starting in the 19th century, including the early archaeological efforts and how cuneiform and the languages that used it were translated.

2. All our versions of Sumerian king list date back to long before the Achaemenid Persians. The oldest is from the reign of Shulgi, an Ur III king who reigned 2084-2037 BCE. Most are Old Babylonian. The best preserved one, the Weld-Blundell Prism, is from the Isin-Larsa dynasty. Later kings would take previous versions and add their own dynasties on the end, giving an impression of continuous unified kingship and the legitimacy that came with it. It's true that the claimed reigns of the prediluvian kings are extremely long, and as are many of those before the kingship comes to Kish, but they become more reasonable as they come closer to the time of Sargon. When it comes to the kings of Ur III and beyond scholars accept the existence of these figures without complaint because we have evidence for them elsewhere, though the narrative of there being only one Sumerian ruler and no overlap for all of this time is less credible.

3. The most common Sumerian claim was that Eridu was the first city. It's pretty hard to know for sure, but there is evidence of a settlement there in the middle of the 6th millennium BCE. There are older Ubaid period settlements like Tell el-'Oueili, but we don't really know how big it was. Uruk was a very prominent early city too It was founded sometime before 5000 BCE and gives its name to the Uruk period (4000-3100 BCE) during which it seems to have exerted a strong cultural influence across southern Mesopotamia.

4. The oldest convincing evidence of writing is in proto-cuneiform from late 4th millennium BCE Uruk. CDLI has an article on proto-cuneiform here. Egyptian hieroglyphics may have been invented independently or inspired by proto-cuneiform, but there's no consensus at this point. As for numbers, while tallies seem to have existed in the palaeolithic, arithmetic systems are first attested in Mesopotamia. Initially there were many different competing systems, with fifteen known from Uruk, but eventually the sexagesimal (base 60) system won out. Stephen Chrisomalis's Numerical Notation: A Comparative History has a chapter that gets into the details of this.

5. We don't have any samples of Sumerian genetic material to study their ancestry from, but the archaeological evidence suggests that they developed from the Ubaid culture, which developed from the Jemdet Nasr and maybe Samarran cultures, and so they were a natural and endemic development of the people who had already been living in Iraq for thousands of years before. At some point their ancestors would have been part of the migration that left Africa some 50-70,000 years ago, but that is long, long before recorded history, and all sorts of movements could have happened in the palaeolithic that we don't know about.

In my opinion, the best introduction to Ancient Mesopotamia is A History of the Ancient Near East, ca 3000-323 BCE by Marc van de Mieroop. If you'd like to start getting into he academic history of Mesopotamia, I recommend starting there!