Did they write things in Genesis?

Little is known about the extent of literacy among the ancient Israelites and other Near Eastern societies. For the most part, only scribes, certain religious and governmental officials, and some wealthy businessmen, along with other elite persons, could read and write beyond the basics. Possibly those with lower socioeconomic standing would have had basic literary training, but the evidence is small.

The invention of writing appears to have occurred in Egypt and Mesopotamia at about the same time—​the late fourth millennium BC—​but neither of those writing systems is alphabetic like ancient Hebrew. 

One must presume that some predecessor of ancient Hebrew, a Northwest Semitic language, is the language the Biblical text refers to in Ex 17:14 since the writing is to be preserved for future reference. 

Ancient Hebrew and most other alphabetic languages (including modern languages such as English) all derive from the same alphabet—​likely a Semitic invention in the first half of the second millennium BC. 

Recent discoveries of primitive alphabetic inscriptions in Egypt have raised the possibility that the alphabet was developed by Semitic peoples living there as early as 1800 BC.

 In assessing literacy rates for ancient Israel, one must consider Biblical and epigraphic evidence, and little assessment can be done for periods before the Iron Age (about 1200 BC). Nothing recognizable as Hebrew appears before this period. From the Late Bronze Age came nearly 400 Amarna letters (fourteenth century BC), written mostly by scribes living in Syria and Palestine. Still, these are in Akkadian (an East Semitic language), though they contain some West Semitic features. 

Apart from those, however, only about 20 texts have survived from Palestine. These date to the general period covered by the Late Bronze Age (1550–1200 BC) and seem to reflect a Northwest Semitic language, but this language is not Hebrew.

 It is not until the Iron Age that actual Hebrew inscriptions turn up. However, this evidence, combined with Biblical references to writing, still must indicate widespread literacy among ancient Israelites. 

There is insufficient evidence for the kind of educational and economic systems that promote literacy growth in human societies. While ancient Israel may have had a higher literacy rate than some other Near Eastern societies—​one estimate for ancient Egypt is a literacy rate of 1 per cent—​only a small percentage of the population could read and write.

Someone with Moses' background would probably have been literate. The Biblical text attributes Moses the ability to read (Ex 24:7) and write (17:11) in a language the Israelites can understand. We do not know what specific language would have been spoken by Israelites coming out of Egypt toward the end of the Late Bronze Age. 

To suppose they spoke some type of Northwest Semitic dialect is not unreasonable. Still, it almost certainly was not the kind of Hebrew we find in the Bible, a dialect of Northwest Semitic that did not fully develop until later. Thus, anything written down in the Late Bronze Age or the early Iron Age and preserved in Biblical texts has been updated for presentation in the kind of classical Hebrew found in Biblical manuscripts.

Keener, C. S., & Walton, J. H., eds. (2016). NIV Cultural Backgrounds Study Bible: Bringing to Life the Ancient World of Scripture (pp. 140–141). Zondervan.