Fixed menu bar, included search form (mobile friendly)

I am using a Child Theme in my Twenty Thirteen theme and I am trying to include the search box in my fixed menu bar (http://188.121.54.43/).

I managed to get the menu bar fixed by using the following code but some style elements are not working yet (for example when you zoom in the menu bar does not fit to the screen. The menu bar is also transparent and it disappears behind the photos in the posts. I would like to fix this too)

\\.menu-home-container {
left: 0;
top: 0;
position: fixed;
padding: 5px 0;
width: 100%;
text-align: left;
background: #545454
}

\\.navigation.main-navigation {
left: 0;
top: 0;
position: fixed;
padding: 5px 0;
width: 100%;
background: #545454;
}

\\#navbar.navbar {
left: 0;
top: 0;
position: fixed;
padding: 5px 0;
width: 100%;
background: #545454;
}

However, the search form is not working anymore. While the search (magnifying glass) icon is still there, it has lost its function.

1) I am not sure what I should change in order to get the search form working. I would also like the CSS code to customise my search form. Is there a CSS code I can include in my style.css to get it working properly? Or some other solution?

2) The menu has one item called ‘All’. Preferably I would want this item to go back to my home page and like to know the code for that. Also, on my mobile phone the menu is called ‘menu’ having a dropdown with ‘all’. I would like to have it appearing in the same way as it appears on my desktop: only showing ‘all’ and no dropdown menu.