Each camera has a list of different settings you can switch between: 100, 200, 400, 800, usually 1600, sometimes one or two more above that, and often another below 100, whether that's 50, 64, or 80. So picking your ISO is something you can do between pictures with the same camera (unless the camera's too automated, in which case this is still happening; it's just that the camera's doing it for you).

Yes.

A wider lens lets more light in to hit the sensor. The more light hits the sensor, the less sensitive the sensor can afford to be and still capture enough light. A less sensitive sensor is less prone to sensor noise. To go for a low-sensitivity, low-noise sensor, there are two things you can do: pick a camera with those traits when buying (if you can; it's nearly impossible to find in compacts, and even Fuji seems to be drifting back into the pack on this now), and pick a low ISO setting when actually shooting.

I take your "focal ratio" to mean "focal length ratio", also known as "aperture" or "f-stop". When the aperture is open wider, more light comes in so it's easier to see more dimly-lit subjects, so ISO doesn't need to be so high. The trade-off is a narrower focus field depth. The depth of field is the range of distances at which things will be in focus, based on distance away from the camera. Whatever the ideal distance away is for focus for a given picture, any deviation from that distance, closer or farther, becomes blurry due to being out of focus. (Sometimes this is done on purpose to avoid letting other things in the picture take attention away from the subject or make the shot feel to cluttered.) It's possible to set the aperture so narrow that everything's always perfectly focused, but then you're letting in less light, so you need to be sure there's plenty anyway or compensate using another setting or light source or such. F-stop numbers work in a strange way, BTW: it's technically
1/x, but only the x is usually reported, so a higher number means a narrower aperture, which means a bigger depth of field.

Lens size is something you choose when buying (or when changing lenses, if you have an SLR, which isn't the kind of camera you're looking at). ISO setting is something you choose by pushing buttons whenever you feel like it. Among compact cameras, there's really practically no variation in lens size. But in general, yes, bigger cameras can fit bigger lenses and sensors and put the lenses farther away from the sensors, all of which is better for image quality, particularly in low light.

Low ISO will give better pictures, as long as there's enough light coming in that the picture can be taken at all, and as long as letting in enough light doesn't force you to an extreme shutter speed or aperture setting which has other unwanted effects (motion blur or really narrow depth of field). As far as I know, all digital cameras have an "auto ISO" setting to let the camera decide, but cameras tend to be too willing to switch ISO up, so I leave mine at the lowest setting all of the time to keep it from doing that. The times you'll want to bump ISO up will be low-light scenes.

Yes. That's one reason why people pay the higher prices for the bigger cameras. (And the high-ISO effect not only adds grain but also dampens colors and flattens shading even when there's no visible grain.) But compact cameras have only tiny differences between each other in terms of the sizes of parts, so the only variable you really have any choice about in the world of compacts is the number of pixels, because fewer pixels on the same sensor means bigger pixels. All I see now, even at Fuji, is 8 or more megapixels. The ones that were getting the praise up to a year ago had about 6, while there were already higher MP counts available; that was a sign of favoring quality over the marketing theme of "more megapixels = better", but now they seem to have joined the "megapixel race" like everyone else. Their electronics might still be better for a given pixel count than others', but I can't trust in that now that they're squishing so many pixels in there; that, plus the bigger screens and lack of viewfinders (another very important thing to me), indicates a general shift in their design philosophy, away from affordable quality and into the established gimmicks of the mass market.

Some digital camera review places might still recommend Fuji compacts over others based on their sensor electronics despite the pixel size now being down to normal, but at those pixel sizes I wouldn't expect it to make much difference, and I'd suspect that such recommendations are simply matters of clinging to tradition. If I were getting a compact camera today, I'd pay attention to some other criteria instead, like waterproofness and/or the presence of a viewfinder.