Elamite was the language of the north coast of the Persian gulf.

http://en.wikipedia.org/wiki/Elam

Some attempts have been made to connect the language to Dravidian, but I think it is a minority view.

I can't speak for Iran, but a lot of research is being done on the pre-IE Harappan civilization of the Indus Valley region in India. I personally have not studied the culture and only have a passing familiarity with it.

Were Dravidian languages formerly spoken throughout North India?

The Harappan language largely remains a mystery (and we can't decipher Harappan script), though some hypothesize that it might have been related to the Dravidian languages. The Dravidian Brahui language is to this day spoken in parts of Pakistan and Afghanistan, and other Dravidian languages are spoken in various small pockets of Northern India. The precise origins of many of these languages is fairly unknown, but many think that they stem from some greater Dravidian language that mostly died out in the North with the arrival of the Indo-Europeans.