Is it possible to change what header image is loaded based upon width? If not, can you make the image resize as opposed to crop when the width of the browser changes?

1) no
2) Not without changing the code. The header area is a fixed height, so say you had it as 250px and 1000px wide. If you had the image resize and made the page width half size, then the image would be 125px X 500px - it would fit the width, but there would be a large blank area in the height since the image area is 250, but the resized image would only be 125.

You could see if there is a widget that will resize and then use the new widget header area and not have a image area...