The BoundingBox looks suspicious to me.
In any event you cannot trivially do what you are trying to do with Ghostscript, because the PDF interpreter uses the information in the PDF file to set the media size.
The first two command lines 'might' work, but you've translated the CTM in the wrong direction. You've moved the origin (0,0) from the bottom left, up and right. That's moved the content of the page further off the media, which is why you get a blank page. You could try using the same values, but negated, so that the origin moves down and left. From the BoundingBox you quoted, that's the correct direction.
gswin32c -sDEVICE=pdfwrite -dFirstPage=1 -dLastPage=1 -o croped.pdf -dDEVICEWIDTHPOINTS=1816 -dDEVICEHEIGHTPOINTS=670 -dFIXEDMEDIA -c "-6292 -6865 translate" -f bbox.pdf
You don't need the rectclip, because the content is already clipped to the page.
The third command line would also work, except that you've set the CropBox before processing the PDF file, so the PDF interpreter reads the CropBox from the PDF file and overwrites the one you set. Try setting it after the input file.
gswin32c.exe -dNOPAUSE -dBATCH -sDEVICE=pdfwrite -sOutputFile=croped.pdf bbox.pdf -c "[/CropBox [6292.907808 6865.505790 8107.091753 7534.493770] /PAGES pdfmark" -f
[EDIT]
OK so the reason the first command lines doesn't work is (as I suspected) because the PDF interpreter resets the graphics state before running the PDF, so it simply throws away the 'translate'.
The second command line works perfectly well for me if you negate the operands in the array for PageOffset:
gswin32c -sDEVICE=pdfwrite -sOutputFile=\temp\out.pdf -dDEVICEWIDTHPOINTS=1815 -dDEVICEHEIGHTPOINTS=670 -dFIXEDMEDIA -c "<</PageOffset [-6292 -6865]>>setpagedevice" -f D:\Users\ken\Downloads\bbox.pdf
The third command line doesn't work because it sets the CropBox for all Pages, which is a default and can be overridden by setting a CropBox on each page. Your original PDF file contains a CropBox (identical to the MediaBox) which is preserved by the PDF interpreter, so the PAGES CropBox is overridden by the CropBox specific to the page.
But the command line above worked fine for me.
