Show special field when correct shipping is chosen

You need to setup the change event on the field:

$('select.shipping-field').on('change', function()) {
   $.post(
       ajaxurl, 
       {
           action: 'shipping_special',
           data: 'your data here..'
       }, 
       function(response){
           // The response should contain your special field HTML
       }
   );
});

You can pass the current shipping select option

Then your theme’s functions.php file needs to have the action to return your special field, if the shipping is zasilkovna

add_action('wp_ajax_shipping_special', 'shipping_special_field' );
add_action('wp_ajax_nopriv_shipping_special', 'shipping_special_field' );

function shipping_special_field() {
    // Return your special field HTML here
}

See wp_ajax__requestaction, this tutorial from WPMUDev or this SO answer;