Not owning a 5DIII I had to download a manual to check this. I would expect that you would potentially see gains to ISO 12800. The ISO values above that though are designated as extended (H) values and would appear to be digitally generated so may as well be done by pushing in camera. At least that is the way that Canon seem to have designated ISO values in other older bodies. The fact that it is digital gain is usually accompanied by a resulting loss of DR. This is quite easy to see when using intermediate ISO values too, which are also pulled or pushed from the closest full value.
The testing that I'd done to compare ISO levels was subjective (i.e. I've not done any mathematical analysis of the results) but from reading up on results from other bodies I think that the analogue gains usually seem to drop off before you get to the extended ISO values.
As you rightly point out, intermediate ISO values are often the result of manipulation from other amplifier settings.
Of course even if the amplification used is analogue there will come a time when using it is not necessarily going to gain you much in the way of additional SNR, but I would always be inclined to keep going with analogue gain, unless it actually made the SNR worse at any given combination of shutter speed and aperture. Also of course you still have the benefit of shooting at the highest ISO value at an ordinary level before you have to start pushing it.
Agreed; though of course remember that by using a higher ISO level you are risking more clipping in the highlights. So though you may not be getting worse shadow detail it may still be better to stay with a lower ISO level in some situations.
For cameras with Sony sensors, and likely many other non-Canon sensors, ETTR really is almost pointless above the base ISO. Yes, pushing the histogram as far to the right as you can without clipping highlights is desirable -- but reducing the ISO is more desirable, if you have the choice.
One other clarification: I'm interpreting ETTR to mean that you're purposely overexposing the scene, relative to how you will ultimately present it. In other words, that you plan to reduce the exposure in software after the fact.
Agreed on your clarification. Note though that there is a reason for using ETTR even when using a Sony sensor; darker pixels will be digitised into smaller numbers, with lighter pixels resulting in larger values.
Imagine you have an image with a sky that darkens by one stop in a smooth gradient from the top to the bottom. If your exposure settings will result in this sky being quite dark in the image on the sensor the pixels will have small values; e.g. 64 at the top and 32 at the bottom (the bottom being roughly half the signal strength, thus one stop darker).
If you instead exposed the scene so that the sky was very bright on the sensor the values may be, say 16384, and 8192 respectively.
When you come to edit this image in post, any attempts to increase the contrast in the sky (essentially increasing the difference between the brightness of the top and the bottom) might result in obvious banding and posterisation because there are only 32 (64-32) steps between the top and the bottom of the sky in the original image.
Doing the same with the brighter exposure gives you far more latitude because there is significantly more tonal gradation.
Obviously that is an extreme example, but even ignoring photon noise and Canon's Shadow Banding(tm) feature, ETTR is very beneficial.
EDIT: An old article, but hopefully useful: http://www.luminous-landscape.com/tutorials/expose-right.shtml