Need dimensions for header graphic and logo

When I went to create the slider for the header, it said the recommended size is 1600 X 400. However, there was huge white space under the graphic at 400 px. I changed it to 555 px and it now it has just a bit on the bottom but space on the top now shows the html coding .

For the Logo, I need to know what size I need. I had one that went across the entire space, and the menu showed up underneath, but that does not work anymore after the updates to wordpress. When I put in a logo, it puts in a black background and centers the logo if I use a small one.

Your help is much appreciated.