You can try below code. Below code shows search box and woocommerce product categories . It will also show selected category.
<form name="myform" method="GET" action="<?php echo esc_url(home_url("https://wordpress.stackexchange.com/")); ?>">
<?php if (class_exists('WooCommerce')) : ?>
<?php
if(isset($_REQUEST['product_cat']) && !empty($_REQUEST['product_cat']))
{
$optsetlect=$_REQUEST['product_cat'];
}
else{
$optsetlect=0;
}
$args = array(
'show_option_all' => esc_html__( 'All Categories', 'woocommerce' ),
'hierarchical' => 1,
'class' => 'cat',
'echo' => 1,
'value_field' => 'slug',
'selected' => $optsetlect
);
$args['taxonomy'] = 'product_cat';
$args['name'] = 'product_cat';
$args['class'] = 'cate-dropdown hidden-xs';
wp_dropdown_categories($args);
?>
<input type="hidden" value="product" name="post_type">
<?php endif; ?>
<input type="text" name="s" class="searchbox" maxlength="128" value="<?php echo get_search_query(); ?>" placeholder="<?php esc_attr_e('Search entire store here...', 'woocommerce'); ?>">
<button type="submit" title="<?php esc_attr_e('Search', 'woocommerce'); ?>" class="search-btn-bg"><span><?php esc_attr_e('Search','woocommerce');?></span></button>
</form>