In Why does intersect not calculate polygon areas? the answer is "The area values in attribute table are just random attributes in the eyes of QGIS. It is therefore perfectly normal that these attribute values are not manipulated when running Intersect.
Intersect is provided by ftools plugin (a Python plugin). You can expand the tool's code to calculate the area of the resulting intersecting areas."
What would be the "code to calculate the intersections' areas" and to export them? It would be ingenious to do all tasks in ONE tool.