Well I can't really answer the second part but as for why few ancient Mesopotamian cities survived into the common era there is no one answer that fits all cities. (Note: I'll talk mainly about the Sumerian cities.)

First and foremost though environmental factors were central to the decline of Sumerian cities. During the late 3rd and early 2nd millennium southern Mesopotamia saw an increasing salinity in their land due to increasingly heavy use leading to declining agricultural returns which in turn led to a significant decline in population. Cities such as Eridu towards the end of their final occupational phases saw the encroachment of dunes and a higher salinity in the water table which made further occupation untenable. Further, the shifting of the Euphrates and Tigris rivers along with the receding of the Persian Gulf played a major role in the decline of cities such as Ur and Uruk. For example when Ur was founded (4000-3800 BCE) it was in the rich marsh lands near the Persian Gulf. By the time of its abandonment (600-500 BCE) the level of the Gulf had fallen by nearly 2.5 meters leaving Ur far from both its position on the coast which had made it economically relevant and the fertile lands that had supported its large population.

This decline in southern Mesopotamia's population was compounded by increasing instability and Ur, the greatest Sumerian city, was sacked by the Elamites in around 2000 BCE (commemorated in the Lament for Ur) an event the city never recovered from. Indeed after the fall of the 3rd Dynasty of Ur (2112-2004 BCE) the Sumerian cities were never again ruled by a native dynasty and increasingly lost political and cultural relevance as their populations continued to decline under the rule of successive foreign powers such as the First Dynasty of Babylon (1830-1531 BCE), the Kissites (1600-1155 BCE), the Neo-Assyrian Empire (911-609 BCE) and so on. New cities founded by these conquerors in more advantages positions contributed to the decline of the remaining Sumerian cities which towards the 1st millennium BCE found themselves in increasingly geographically irrelevant positions due to the shifting of the two great rivers.

Continuing on the political instability train of thought many of the Sumerian cites over the millennia were damaged or outright destroyed by war. As mentioned above Ur was one of them but also Kuara, destroyed by Sargon II in 709 BCE, Mari (a Mesopotamian but not Sumerian city) was nearly completely annihilated by the Babylonians and continued to exist as only a small settlement after 1759 BCE and Babylon herself was marred by constant conflict till its final abandonment in 141 BCE in favor of Seleucia founded by Seleucus I Nicator in 305 BCE.

Of course some of the great Sumerian cities like Shuruppak were destroyed by fire and others like Akkad we have very little idea of what happened other than they disappeared.

I think it important to remember though that many of these Sumerian and Mesopotamian cities lasted through many millennia. At the time of its final abandonment Ur was nearly 3300 years old and Uruk was perhaps over 5000 years old. These cities weathered geographic and environmental shifts, wars, massive political upheaval, fires and plagues. That some of them lasted as long as they did is pretty damn impressive.