Just to demonstrate a point. The Canon image in the samples above looks quite terrible. While you can probably never extract the same amount of shadow detail as the D800 with a 5D III, it need not look quite that bad. Here is a copy that was run through both Topaz DeNoise 5, with vertical debanding, 100% chroma noise removal, and minor luma noise removal, as well as maximum dynamic range recovery. It was then run through Nik Dfine 2 in order to remove more luma noise. DeNoise 5 does an excellent job removing chroma and banding noise, while Dfine 2 does a better job at reducing luma noise without smudging detail.
Here is the result:
Clear loss of detail on the upper beam relative to the D800. Still some noise on the door frame and window relative to the D800. Some artifacts on the no smoking sign (although more likely due to the use of a JPEG as my source...denoising from an original RAW or high quality TIFF would produce better results.) Despite those, the results are FAR more acceptable than the original 5D III RAW without any noise reduction. Dynamic range has improved considerably with some effective NR. Not the same 13.2 stops as the D800 has, but definitely more than the 10.97 stops that DXOMark measured.
So, yes, the 5D III, like most other Canon sensors, has some pretty crappy shadow noise. Doesn't mean the buck stops there, however.