Is it possible to re-configure an existing widget? I would like to change the functions of the analysis > extract to data widget. Instead of extracting/downloading (save results in) into my "contents" in ArcGIS Online, I prefer to extract/download the data to my desktop.
Yes, you should be able to. Make sure you're in edit application, then at the top left, there should be a list of buttons, of which you'd click widget. Hover over the widget you want to edit and in the lower right corner of the widget's icon, you should see a pencil. Click the pencil and it will open the editing page for the widget.