I would like to add also an “icon” (fa-calender) to the nav. button. Unfortunately i did not understand how this works in the docs! Could you please help me with this issue?

Additional question:
i would like to change the color of the menu in the “sub header” while transparency mode. How can i do it?

Thx & best regards,
Adi

The styling of the header nav button is in your main theme CSS file – style.css, you could override that in your Child theme. For example something like:

#menu > ul > li.boc_nav_button a {
    background: rgb(44, 168, 255);
    border: 2px solid transparent;
    color: #fff;
}
#menu > ul > li.boc_nav_button a:hover {
    background: #444!important;
}

As for adding an icon to it, in your Appearance->Menus, where you actually added your button, simply add the button in front of the button label like so:

<i class="icon icon-calendar-o"></i> YOUR BUTTON LABEL