What has been learned about the Sumerians since the 1960s?

First, a clarification: the translation of cuneiform documents began over a hundred years earlier than 1963. Scholars have been able to read cuneiform documents since 1850, and most of the crucially important documents were already translated and being discussed nearly a hundred years ago. Assyriology is a very new discipline compared to Biblical Studies or Classics, but it is an established field of study; major school textbooks have benefited from the advances in our knowledge of cuneiform since at least the 1920s.

What has been learned about the Sumerians since the 1960s?

The short answer to your primary question is that, since 1963, we have learned basically everything and nothing about the Sumerians. One the one hand, we haven't learned anything that has profoundly modified our overall picture of the earliest periods of Mesopotamian history (e.g. no new cultures, languages, invasions, or major historical revisions), but on the other hand, we have made tremendous inroads into our understanding of the basic outlines of many important aspects of “Sumerian” society (e.g. religion, law, domestic life, some economics, education). Some have reasonably argued that Sumerian wasn't “really” deciphered until the 1980s (e.g. Attinger), and still others would say that it isn't fully deciphered yet today because we do not understand all the intricacies of the verbal system. In any case, we do know quite a bit more about the earliest literate periods of Mesopotamian history today than we knew in 1963.

A full summary of all the advances made since the 1960s is way beyond what I can give in a Reddit post. I recommend taking a look at a recent history book (check our reading list) and compare it with Kramer. You'll see a multitude of improvements in the details, especially concerning society and culture.

Probably the biggest shift since the 1960s has been the supplement of primarily archaeological information with historical documents. I'm grossly over-generalizing, but most accounts of the “Sumerian” times back then were based on a few key texts that were situated in a wealth of archaeological theory and data. Today, we have access to many more texts and archives broadly, which allows us to situate events in a more detailed milieu. One example of this is that we now know the Third Dynasty of Ur (“Sumerian Renaissance”) period relied on war and the labor of prisoners of war by a close examination of their rich economic archives, but our earlier histories largely ignored the role of war since they left few monumental inscriptions describing conquest. This is just one example, but there many like this.

Two key advances have significantly changed our understanding, and they are related. Starting in the 1970s, our knowledge of the city of Mari, in eastern Syria, drastically improved our understanding of Sumerian education (and a ton of other important stuff from later periods like the Old Babylonian). This was further reinforced and magnified in the 1980s with the discovery and publication of the city of Ebla, in western Syria. This city had an impressive archive of tablets, but unlike Mari, most of these were in Sumerian or Eblaite, which uses Sumerian signs. Taken together, the research into these two sites drastically broadened our picture of the reach and horizon of “Sumerian” culture far into Syria from southern Iraq. We now know that a scribal education in Sumerian cuneiform was the model for literacy in the third millennium throughout Iraq and Syria, and we have a fairly good idea how they studied it and why.

For more general information see the reading list or Postgate, J N. Early Mesopotamia: Society and Economy at the Dawn of History. London: Routledge, 1992. Print.

Is it now established that Dilmun is in Bahrain?

Generally, yes. Dilmun is almost certainly Bahrain. There is still uncertainty about the whether it is only Bahrain or a set of lands in and around the Gulf, but the link between the two is accepted now.

See Crawford, Harriet E. W. Dilmun and Its Gulf Neighbours. Cambridge, U.K: Cambridge University Press, 1998. Print.

Is the Sumer-Shem connection more established/ruled out?

Totally ruled out. Poebel's hypothesis proved to be totally untenable once we had a better understanding of various names and their periods of use. We still don't know the source or inspiration for the Akkadian term shumer, but it's almost certainly not connected to the name or people of Shem.

See K. van der Toorn, P. W. van der Horst (Jan 1990). "Nimrod before and after the Bible". The Harvard Theological Review 83 (1): 1–29