I see that the Canon 7D has a pixel resolution of only 2.1 megapixels for 1080p Full HD mode, 0.9 megapixels in 720p HD mode, and just 0.3 megapixels in VGA mode.

Since the sensor is 18 MP, how does the sensor generate these lower resolution images?

The image on the rear LCD does not look smaller or more cropped than the image for a still photo.

When I shuttle back and forth between 1080p and 720p, the image size does not change.

It skips lines. I believe it uses every third line on the 5D, not certain about the 7D - but it's part of what causes the aliasing problems.

As Evan said, it leaves out alot of the data when reasing the sensor. Because it is not possible to read out the full sensor 24 (or 60) times a second and down-convert on the fly, the camera only reads out some of the lines to put together the image in video mode. Because big chunks are missing from the image, we end up with aliasing.

The only cropping you will see in video mode is to remove the bottom and top parts of the 3:2 image to get a 16:9 frame - unless you are using the 550d which I believe has a standard-def crop mode that only uses the central part of the image.