As a certain kayak coach once told me, there are no absolutes :-)
Why would I choose f4.5 as opposed to f5 or f5.6 or 6.3 etc? Would it solely be down to whether I am shooting an action shot or not?
The answer is, it depends.
Lenses are usually at their sharpest around f/4 to f/8 with the sweet spot usually being around f/5.6. In real world shooting those smaller adjustments just mean that you have more fine adjustment. On a fully manual lens video camera the iris is totally analogue with infinite variation. Most stills cameras adjust their iris settings in set clicks, so sometimes you will need to balance between the settings with some variation here and there.
If you make a decision to use f/5.6 because of ideal lens sharpness and have the camera on iris priority, you can pretty much not worry about whether to use f/4 or f/6.3 etc. In other words don't be too concerned about it.
2) If I was photographing an action shot, I take it I would set the camera on the shutter priority, but do I choose 1/4000, 1/3200, ..., 1/200 etc? Would that be simply down to the ambient light?
Beyond a certain amount, usually around 1/1000th, maybe less, any increase in shutter speed isn't usually worth it for anything other than the very fastest of objects.
Lets say that you had the camera on shutter priority and had set it to 1/1000th to capture fast kayaking action. But you found that even with the camera automatically adjusting all the other settings the sunlight was just too strong. The camera may also, in bright light in shutter priority mode, be adjusting the iris right down to f/16 or even f/22 where diffraction becomes an issue (in other words the picture becomes noticeably softer at those settings depending on the lens that you are using).
Likewise if you set the shutter speed too high in dingy light the camera may have to increase the ISO to levels that are really too grainy. Most modern cameras are low noise, but it can still be an issue on some older models. So in this case you may need to lower the shutter speed.
To stop this from happening, and to help the camera get a good exposure, you may need to increase or lower the shutter speed, even though it may not offer any real benefit to the photo. In full manual mode adjusting the speed to be quicker just gives you more options for balancing the other settings.
3) If I'm not going for an action shot, or a narrow depth of focus, which settings should I be choosing first - aperture or shutter speed? And why should I choose one over the other?
Again there is no right or wrong answer/reasoning. In such a situation you need to be choosing the settings based upon how you want the photo to look. For example for a landscape you might decide to use f/11 for the best compromise between lens sharpness and obtaining deep depth of field. Then you could set your ISO to a minimum to make sure that noise is not an issue at all. You would then adjust your final exposure using the shutter speed. The process is vastly different in video however because you should not adjust the exposure with the shutter unless you want the video to look extremely amateur.
So it all depends on what you want to get from the final photo. Experimentation is the key.
I would highly recommend messing around with the full manual settings in order to understand what the camera needs to do and what it does when you use priority modes.