Set maximum header width, post & page missing header

I can’t seem to set a maximum width for the header in the Swiss theme, how can I do that? I have a 1600px wide image, and don’t want it stretched. I tried adding:
max-width: 1600px !important;
to the stylesheet, but it didn’t help.

Also, the header doesn’t show on the posts & pages. I’ve never seen that with other themes. I’d like to have the header show everywhere.

One other thing: with a small mobile screen, the word “Menu” appears on the menu bar, pushing other menu items down, thus creating a thicker bar. How can I stop that from happening?

Thanks for any help.