Solution for this topic

My first question is, there is a way to move the Search icon next to the main menu navigation bar? Like, next to “Galleria” in the picture I have attached. For the moment, I have hidden the Social and Bookmarks icons, and moved the Search icon on the right using this code:

You can achieve it by adding the following CSS code in the Custom CSS code option of your theme on the below path.

Admin Area -> Appearance -> Theme Settings -> Custom Code -> CSS Code

li.menu-item.sh-nav-search.sh-nav-special {
    position: absolute;
    right: 125px;
    top: 75px;
    z-index: 99999;
}

For more accurate results you can use below plugin to display search in the menu.

https://wordpress.org/plugins/add-search-to-menu/

2) My second questions is, how can I use a much larger logo? If I use one with 250px height, it gets stretched awfully.

You can increase logo height using below CSS code.

.header-logo img {
    height: auto !important;
    max-height: 100% !important;
}

Third and last question: can I get rid of the social icons line in the mobile menu?

Use below CSS code to remove it.

.header-mobile-social-media {
    display: none;
}

Solution for this topic

1) The problem is, when the window is down-scaled, the Search icon goes over the menu text (see attached screenshot). I guess I can’t demand too much, but this isn’t the best solution. I’ve tried the add-search-to-menu plugin, but if I use it, then there are two Search functions in the mobile menu…

Using add-search-to-menu plugin is a best solution here and we can remove duplicate search on mobile also so use it on your site and let me know so that I can help you.

2) There are a few problems. If I put max-height (or width) percentages, the logo image is not centered anymore (see the attached picture). If I put pixels instead of percentages, the image is centered, but it is not responsive anymore.

Then please try changing above shared CSS code as following.

.header-logo img {
    height: auto !important;
    max-height: 100% !important;
    max-width: 100% !important;
}

Have you got any news about Gillion implementing Revolution Slider as header? ? This would solve all my problems, also because I would like a big and full-of-effects header in the homepage only, and just a small logo in all other pages.

Sorry but currently we don’t have any ETA.