Will there be any differences between the images captured by the second and third drawing?
Yes. The angle of view is different. You may capture the same amount of the subject in both images, but because the full frame sensor has a wider angle of view it will capture more of the background. To see this, redraw each of the two drawings and add a large object some distance behind the subject. Extend the rays from the lens all the way to the background and you'll see that the full frame sensor captures more of the background object.
You get the same effect with two lenses of different focal length on the same sized sensor. Say you have a 50mm lens and an 80mm lens. You can capture the same amount of the subject with both lenses by moving the camera closer when using the 50mm lens. But the 50mm lens still has a larger angle of view, so will capture more of the background.
If so, are there circumstances where the crop sensor is able to capture an image that the full frame sensor is unable to do?
If you're sticking with the same lens, then yes, the two sensors will see the world a bit differently through that lens. If you're willing to use different lenses then you should be able to get similar photos with both cameras.
Will the zoom of a zoom lens be of a different quality than the zooming contributed to the crop sensor?
This is really what the crop factor is all about. A 50mm lens on a camera with a 1.6x crop factor gives you an angle of view similar to a 80mm lens on a full frame camera because
50 * 1.6 = 80. Conversely, a 50mm lens on a full frame camera gives you an angle of view similar to that of a 30mm lens on the 1.6x crop sensor, because
50 / 1.6 = 31.25 and 31.25 is close enough to 30 that nobody is going to argue.