When you save a page (that shows all linked files fine) from the root directory as a template, it will save itself to the 'Templates' folder in the root directory of your site, and will ask if you want to update the links to which you should answer 'yes'.
Creating a new page (File > New > Page from Template) once saved to any location, should save with all paths to external files updated correctly. The paths in the template are taken into account.
If a page made from a template has somehow lost track of the background image, and you can still see the image in your template, then you may need to re-create or correct the path to the image in your normal page.
If the image is in an area locked by the template so you can't edit it, you may need to 'Detach' the page from the template: Dreamweaver Menu > Modify > Templates > Detach from Template.