Great suggestion, this is possible but would have to be a custom widget.
For exporting, I believe you would need the following workflow:
1. Create a copy of the source code from the select widget
2. Within the source code, add an option to the drop down buttons to export to CAD
3. Use a polygon feature service that has all of your projections you would like the data to be in (ie feet, meters, UTC etc.).
4. Get the center point of the data the user selected and has chosen to export.
5. Use that center point to select by location on the projections feature service to get the desired projection.
6. Reproject the data into the new projection.
7. Export to shapefile or some other CAD data (the default within the widget is geojson, not sure how to do this part, conversions to CAD data can be difficult, shapefile easier)
AutoCAD can take in shapefiles, so this may be the easiest route.
Translating DWG into GIS data can be very difficult, and I don't have a recommendation on how to do that.