Jeff - Not sure if you've ever seen this but the 1936 Chevrolet Service News Bulletins has a detailed listing of the
https://chevy.oldcarmanualproject.com/bulletins/36csn58.htm for cars and trucks. Nothing in there directly mentions an offered flat bed for the GC 1/2 ton or the auxiliary springs for the GC either. But there is a lot of info to supplement/support the GM Heritage files
1937 Chevrolet Truck PDF.
Although the '37 PDF doesn't display a drawing of the GC with cab (Model ID - CCAB), the
1938 Chevrolet Truck PDF does show a drawing of the "chassis with cab" (also designated "CCAB"). The screen grab below comes form the '38 PDF and clearly shows the model including running boards and rear fenders, so it likely IMO that the '37 GC came similarly equipped.
I think the only way to perhaps gain more definition to these questions could come from a 1937 Chevrolet Truck Salesman's Data book. If you're interested enough to dive after more details, you can order a reprint of a one from Jim Carter as
LI189. Lots of sellers offering them, but Jim Carter has the best price I could find.