To convert GIS data to CAD in ArcGIS:

In ArcMap, click on ArcToolbox and go to Conversion Tools.  Select To CAD, then Export to CAD.  Select the layers you would like to convert and the type of file (based on your preference or the version of AutoCAD you are running).

To clip out an area of interest:

Zoom in on your desired area.  Right click on the layer you would like to clip, and select Data, then Export Data.  Select "All features in View Extent" from the drop down. When asked if you would like to add that to the map, click Yes to view your new data layer.  You can then follow the steps outlined above to convert this to CAD.