When searching for any item in the store, it does not return any results. However, in the search field on the next screen, the results appear normally.
If the width is less than 991px it works OK.
I have found that this is because there are two sets of code for search on the page. The first one is only enabled when width is <992px.
This is caused by an error with line 57 of common.js not reading the correct search input field.
57 var value = $('header #search input[name='search']').val();
It always finds the the first instance of #search even when the second instance is active(>992px)

I have been able to fix this on my installation by editing the common/menu twig using the Theme Editor. Removing the second use of {{search}} and {{cart}} and moving the position of the first ones.
Resulting code for common/search.twig attached;
menufixed.twig.txt

; Seems to work well on all widths responsive.

Hello. From what i see on your demo you have a problem with cart icons and search icon on mobile... that seams that are not responsible :)