Inserting and placing an image

I want an image between the header top menu and the primary menu with my site name and tagline on top. I can add the name & tagline to the image if necessary. I've tried to follow the instructions for a header image, here's the problems I have. 1. The original image size is 3260x1780 px, well within the 1200 x 400 recommended. It doesn't cover the full width of the page. 2. I can't seem to get rid of the search box. (I've gone through all the customize options.) This leaves white space above or below the image depending on where I place it. Am I getting this all wrong? Should I be trying to do something else than set a header image? Is there code I should be inserting somewhere in the CSS? I can copy and paste if I know where to put it. I'd appreciate any suggestions. Thanks.