I need to make my site 1024 px wide

I want to get rid of the Header entirely so that the image is the header.

You can use a layout element to do so: https://docs.generatepress.com/article/layout-element-overview/

The Content block below the header does not take up the 1024 pixels to be even with the image.

It's because there're paddings around the content container. You can check the value at customizer > layout > content container.

I would recommend use generate_inside_site_container hook for the image, give it a try.