It all comes down to whether the lower resolution images were created by downsizing the full resolution original and how it is downsized (ie, what type of
digital image interpolation is used).
If the 3 MP is created using
bicubic image interpolation from the 9 MP image, it will have much less noise than the 9 MP image when viewed at 100% on the monitor (for the same ISO/ASA setting). This is because each pixel in the lower resolution image is created based on several pixels from the higher resolution image-- producing a
noise reduction effect similar to image averaging. This applies for many current SLR cameras, because they use the higher resolution original to create a downsized image, and use high quality interpolation to perform this downsizing.
On the other hand, some cameras do not create their downsized images from the full resolution original, but instead just discard the extra pixels, or use something similar to
nearest neighbor to downsize the image. In this case all images would all have the same amount of noise when viewed at 100% on the screen, regardless of the number of megapixels. I will assume that this is not the case.
When viewed on the computer's LCD screen at the same size (such as showing only 3MP), then it comes down to whether your image viewing program performs interpolation on the 9 MP image to convert it into a 3MP image for viewing. I know that photoshop does not perform bicubic downsizing for image display, so the 3 MP image may indeed look better than the 9 MP image at the same ISO setting (unless you downsize the 9 MP image within photoshop yourself). However, this does not mean that the lower resolution image will look better when printed.
The higher resolution image will inevitably be downsized by your printing software or by the printer itself, which is often performed in such a way that will make both images look identical in a print.
Another thing to consider is the image sharpening.
Downsizing an image can introduce some softening of the photo-- requiring sharpening after the downsizing has occurred. I am sure that the camera performs this on both the 3 MP and 9 MP images, but if you shrink the 9 MP image on your computer and forget to sharpen it afterwards, it will look softer than the 3 MP image.
Overall though,
the highest resolution image (9 MP or case 3 from your post) will always be the safest bet and give the best results (assuming that you remember to sharpen after downsizing the photo to your intended printing size). Depending on the camera, the 7 and 3 MP versions may also give identical quality if you do not print the image any larger than the size (in inches) produced at 300 PPI using 3 megapixels.
Also note that
when viewed at 100% on-screen, progressively higher image resolutions will look worse and worse because there are diminishing returns with a camera's optics. This does not, however, mean that the higher resolution image is necessarily worse-- just that you are getting less quality per pixel.
A 10 MP image will still be better than a 5 MP image from the same camera-- just not fully 2X better (assuming the same photo settings)