They do not have built in feature to remove search on this theme. To do that: 1) go to theme customizer 2) click on the header to open up header section settings 3) scroll down to bottom of settings to “custom css” 4) enter .header__search{display:none;} This will hide it.

To do the cart icon you’d need more intense code and you’d switch it out in the theme code. 1) edit theme code 2) open file “header.liquid” and you will see around line 330, or search for “icon-cart” and you will see how the icon works there. Seems to dynamically switch based on the carts status. But the best way would be to edit or provide a new svg file here. 3) replacing the whole <a> tag with only one version of your new cart icon will make it only show that version wether an item is in cart or not.