Of course they don't have to take up the entire width but specifically as a designer, my opinion is why wouldn't you? Theres so much real estate to be used in the entire width with larger monitors why would you want to limit yourself to 960px? I personally am not really a fan of the dominos site and if they hired me, I'd definitely use the full width if they let me. But if you notice it is responsive and changes to full width as you change the width of the window. This is all just my opinion though, by all means, if thats what your client is into, go for it. If you look at the majority of the pages on https://onepagelove.com they are all full width with beautiful hero images. That's the site I go to for inspiration from award winning designs.
But to answer your original question, What I did to get that effect in my screenshot was to put the logo back into the logo container in the navbar and give it a negative lower margin. That makes the navbar go back to it's original height. Z index won't help in this case. Let me know if that makes sense, if not, I'll make you a video.