Hello,
First of all, you won’t lose the custom CSS that you add to Theme Options > Custom CSS after the parent theme update. If you switch theme to other than XStore this option won’t be available at all because this is the XStore theme option.
If you want to hide the mobile menu from header add the below code
.mobile-header-wrapper .et_b_header-mobile-menu .et_b-icon {
display: none !important;
}
3) Clear the browser cache or check using incognito window. Because I tested your site using mobile device and don’t see any problem with the mobile menu or more buttons.
Regards