This is my understanding of how things work:
Reporting TCs: A TC-compatible lens (e.g. 400/5.6) detects the presence of a TC through the extra contacts and reports the correct f-stop for the combination to the camera.
Non-reporting TCs, or taped TCs, or TCs with non-TC-compatible lenses: The lens does not detect the TC and just reports its own f-stop. AF may still work, because the camera doesn't know about the true f-stop, and tries to AF even if the combination is outside the specs.
Kenko DGX: Unlike the above TCs, the DGX manipulates the communication between the lens and the body so that the f-stop is always reported correctly for the combination, and so that AF is retained even if the combination is above f/5.6. From your and others observations that f/8 is reported but AF still works, my guess is that wide-open f-stop and used f-stop is reported separately and the camera looks at the wide-open f-stop to determine if it should try to AF.
Personally unless you have fantastic light conditions a 2X TC on either of your lenses won't produce great results. Nikon's slightly different to Canon, Canon limit autofocus to f5.6 for all but there 1D series camera's, with Nikon autofocus still works (I've been led to believe) but with both brands it depends on the compatibility of the lens with the TC as only certain lenses work.
TCs are really designed for the faster prime lenses not zooms because the optics can handle the drop off in image quality and hit on the auto focusing, but you can use a zoom and TC but be aware you'll need decent light conditions to get anything worth while image wise.
Personally I would never even think about attaching a TC to a 70-300mm lens