There are researchers who link ancient Sumerian to a now unknown Dravidian predecessor language, as well as to the Sa'ami and Hungarian languages (in part).
Both Sa'amiand Hungarian/Estonian have been overwhelmed ~ almost swamped in Suomi (Finnish) vocabulary though.
Archaeologists working in Iran have managed to uncover an ancient city that may link North Eastern India (the original Dravidian area) to Western Asia ~ this one is quite a bit older than the much more recent ruins at Mohenjo Daro. So far all this discovery does is demonstrate that migration was not a one way street with bronzed blond Aryan supermen raging East over the North Indian plains, but seems to have started with short dumpy guys with cows humping it over the Great Sind desert to Mesopotamia and beyond!
Sumerian, of course, really has its roots in the accounting necessary to keep track of cattle!
One of the great linguistical mysteries has been the origins of the Basque language. Is it starting to fit into the Dravidian sources you reference?
True about cuneiform - the oldest written document is said to be an inventory list or receipt IIRC.