If I shoot in manual I set the exposure to the nearest 1/3 stop. If I use one of the automatic modes does the camera do the same thing or can it set the dependent variable to a resolution finer than 1/3 stop?
It can depend upon the camera and lens you are using, but the vast majority of cameras use the same step sizes whether shooting in manual mode, a semi-automatic mode, or fully automatic mode.
In manual, cameras have 1/2 and 1/3 stop increments because it is difficult to discern 1/3 of a stop, much less a finer resolution than that, and smaller increments would simply mean having to spin a thumb wheel that many more times to get to the setting you want.
In an auto mode you would think the camera could freely choose any intermediate value it wanted.
However, with ISO, best results often come at specific multiples of the base ISO, so finer tuning might yield poor results.
And there is no doubt a limit to how finely the camera can control the lens aperture.
Depending on shutter design, I would expect the camera could achieve shutter speeds finer than 1/3 stop increments.
But in the end the camera has to calculate exposure, write EXIF data, and display the aperture, shutter speed and ISO in the camera's displays, and it may just simply the onboard processing to keep things consistent to how they are in manual mode, in those discrete 1/3 stops.
In a practical sense, it really doesn't matter because you really won't be able to tell, even in a precisely controlled experiment, for example a 1/6 stop difference. You probably can't tell a 1/3 stop difference.
For usual photography (i.e. not high precision scientific measurements), there is no point in setting it in finer increments. The mechanical shutters and mechanical apertures used in these cameras are simply not that accurate. You can try this by firing off a few shots in continuous mode, and looking at the histogram of each shot. Set your camera to display the histogram in playback mode, and flip through the images. You will see obvious variations between the shots.
Yes they can and many but not all do. It depends on the camera model but often cameras with manual controls use the same step sizes which makes it simple to switch between the automatic and manual exposuse and get the same results.
On digital sensors ISO can almost be continuously variable but some sensors have dedicated circuits to provide better gain and a cleaner signal at some multiples of the native sensitivity. This is why some cameras have better performance at ISO 800 than 640, for example. Yet some cameras do it. You can check EXIF to see what happens, I have seen some ISO 158 shots!
Shutter-speeds are also easily divisible in small increments and many cameras take advantage of that to compensate for a lack of aperture or simply because they can. On the Fuji X-Pro2 for example that I was shooting with on the weekend, metering at ISO 200 and F/5.6 gave a shutter-speed of 1.1s which is not possible to select manually, so I settled for 1s which is close enough and reduces the risk from details burning out.
The most restrictive component is the lens. Not all have a continuously variable aperture, so they may have set stops built-in, in 1/3 or 1/2 increments usually. There are continuously variable ones which are advertized as better for video as they create a smooth transition when a scene changes. I guess that electronic aperture control has a set number of bits to select aperture so that might also add yet another limit to increments.