The genesis of a language, assuming its organic nature, is something that is still hotly debated among evolutionary biologists, cognitive psychologists, and linguists alike. Conlangs (constructed languages made by individuals or groups) did not really come about until much later in history (medieval times really, like Enochian). I have seen no compelling evidence that Sumerian or any of the other ancient mesopotamian languages were conlangs, so it is unlikely that a single individual "came up" with a language here and disseminated it throughout society. Beyond this, I cannot comment on the linguistic evolution of Sumerian, Akkadian, or Aramaic.

I can, however, comment on the way that writing was taught in the Fertile Crescent. The Sumerians had initially developed writing in the city-state of Uruk, where cuneiform evolved past its proto-cuneiform stage. We know that Uruk is the birthplace of human writing because the oldest cuneiform documents can be found there, and the very Sumerian symbol for "city" was the same as that used for "Uruk" (compare other city-states whose names are differentiated). The ancient Sumerians had a system of schools called edubba designed to train scribes (dubsar). Writing in Sumeria developed not for religious purposes (at least initially, as it would certainly attain a religious element later on), but rather for accounting purposes; early transactions had merchants send clay envelopes filled with tokens alongside their merchandise to their customers. The tokens would correspond to the type and quantity of goods to ensure that the correct goods were traded, and pictographs detailing the contents of the envelope were stamped on the outside to ensure that the customer knew if goods and their tokens were stolen. Eventually, they did away with the tokens altogether and sent along clay tablets with pictographs instead. The pictographs became more abstract as time went on, resulting in wedge-writing we know today as cuneiform.

Scribal edubba changed in format over time, and by the Third Dynasty of Ur, schooling was done in a master scribe's home, where his students (dumu edubba, "children of the school") were instructed in a wide range of topics, including arithmetic, history, religion, and wisdom literature (proverbs, catchy sayings). There was even a recess period and teacher's aides (shesh'gal) to watch over students during breaks.

Regarding language, the Ancient Sumerians spoke Sumerian, a language isolate not definitively know to be related to any other language. Their successors, the Akkadians, spoke Akkadian, a Semitic language related to Aramaic, Hebrew, and Arabic. When Sargon of Akkad forged one of the first hegemonic powers of the Middle East, his people adopted the Sumerian cuneiform to write their own records. This led to some difficulties, as cuneiform was tailored to the structure of the Sumerian language, not Akkadian, so various methods had to be employed to allow Akkadian to be represented well (such as the use of rebus, that is, using homophones, i.e. Eye-sea-ewe standing for "I see you"). The Akkadians would actually retain the use of the Sumerian language for religious and scientific purposes, much as how Learned Latin was used in Europe or Sanskrit in India, Classical Chinese in China, etc. It was a sign of prestige (and pretention) for an Akkadian scribe to be able to write and speak in Sumerian, and as such Sumerian was often taught in their scribal schools.

If you'd like to know more, here's a writeup I did a while ago addressing the structure of the Sumerian education system.