There are actually several possible answers to your first question- when Sumerian died out as a spoken language, when Sumerian history and culture were forgotten within the cuneiform tradition(if they were) and when cuneiform culture as a whole died out. I'll focus in this post on addressing the first and last points because they involve the relatively clear problems of dating and defining language death.

The timing of the first event is highly uncertain; we can't get very far on the evidence of literary production because the bulk of our Sumerian literary texts are A) school texts and B) written after Sumerian probably died out as a spoken language. We can make some estimates based on personal names and hints in texts that are securely dateable to periods when it is reasonably certain Sumerian was still spoken. The general consensus is that Sumerian died out about a century or two either side c. 2000 BCE, although people have argued for dates more or less recent(as recent as 1650 in perhaps a tiny handful of isolated pockets) and that at some point people spoke it as a native language.

Another possible answer might be with the final death of the cuneiform tradition. After all, Sumerian lived on as a language of scholarly life and especially of cultic and astronomical texts long after it died out as a spoken language in virtually all plausible contexts and within an otherwise Akkadian cuneiform culture and as late as the 7th century the Assyrian monarch Assurbanipal bragged that he could read Sumerian. Other aspects of Mesopotamian culture also survived with this use of Sumerian such the earliest pantheon(even if Marduk had replaced Enlil as head of the gods), some kinds of texts like the Lexical lists used for scribal education(and some lexical lists such as Urra, Ea and Lu existed almost continuously from the archaic to the Neo-Babylonian period), and stories like those of Gilgamesh. If we take that as an answer, we would say that Sumerian civilization was mostly lost when cuneiform first became essentially the preserve of a handful of priestly families mostly from Uruk during the Seleucid period after Alexander the Great's death (probably around the middle of the first century BCE) and then entirely died out.

Now some parts of the Sumerian and Akkadian corpus were transmitted to the classical world. Berossus's Babyloniaca, written by a priest of Marduk in Greek, has survived in fragments up to the present and exhibits clear familiarity with Sumero-Akkadian historical and mythical literature(although the surviving fragments do not speak of Sumer or Sumerians, not even in the reflexive way that cuneiform first-millennium texts call Babylonia "The Land of Sumer and Akkad") , and it is possible that the Alexander mythos and some Islamic legends contain faint echos of Mesopotamian mythology. But on the whole, it would be fair to say that at this point the cuneiform tradition became basically inaccessible and would not be rediscovered until the decipherment of cuneiform thanks to the Persian royal trilinguals and much more importantly the discovery of Assurbanipal's library at Nineveh and several other massive archives of tablets in the mid 19th century.

Addendum, since it is somewhat implied by your question specifying "Sumerian" civilization: We don't actually have a very clear handle on who the Sumerians actually were, not least because they are not very clearly distinguished in surviving texts as an ethnic group. There are a few scattered mentions of people being men of Akkad or Sumer and claiming to be "a son of Sumer, from it seed" dating to the Ur III period, but nothing very clear and the Sumerian school texts that as I mentioned earlier are our largest corpus of Sumerian seem to mostly presume bilingualism. There is intriguingly one first-millennium text in which "Sumerian" is used "to qualify the owner of some Nippur tablets or his ancestors"(Cooper 2012) , which is especially interesting since Nippur during the Old Babylonian period was notably a very "Sumerian" city and had a high level of Sumerian scholarship. Basically, the situation with Sumerians and Akkadian as distinct ethnic groups within the Babylonian world(as opposed to the idea of being Babylonian rather than Amorite, Elamite, Shubrian, or any number of things, which is well attested) is that there probably was some idea of being a Sumerian but what that actually meant or how it was defined remains entirely obscure.