Ok, for the header background, take a look around line 2536 in your style.css and either Remove, Comment out….or change the color of this code –
.front-page .site-header {
background-color: #000;
}
This shows up in the responsive section to override the main menu when the site is being viewed on mobile devices. So if you remove or change the color then it will show up as the blue color you have.
It wasn’t originally designed to stay fixed on mobile devices, so if you want it to be, you can start by editing line 2532 in your style.css and Remove or Comment out this code –
.site-header {
position: static;
}
That will revert to the default position: fixed;
which should make it show up in a fixed position. You may have to make additional adjustments to the logo and menu area if you need it to show up a certain way.