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Hi,

Could you point me to the GIS site where you downloaded ZIP archive?
I think the main reason of the errors you get in AutoCAD is that you are confused with file formats.
AutoCAD uses its own SHP and SHX file formats - you can define shapes to use as drawing symbols and text font:
http://docs.autodesk.com/ACDMAC/2013/ENU/filesACGMac/GUID-922036EE-45E3-4808-AA9D-663240E92E6D.htm

But you downloaded ESRI Shapefiles - digital vector storage format for storing geometric location and associated attribute information.
That is why you was not able tp use AutoCADs COMPILE command and cannot load SHX file to AutoCAD - the files you downloaded are in another format which plain AutoCAD cannot handle.

Look also here:
https://forums.autodesk.com/t5/autocad-2013-2014-2015-2016/importing-a-shape-file-from-arcgis-into-a...

Maxim