Custom Icon and Class Selection

You will most likely need a custom formatter that extends from LinkFormatter

In the :viewElements method you have access to $url - and on that $url->getOption('attributes') will be the attributes, you can adapt them there and then call setOption

Something like this:

namespace Drupal\YOURMODULE\Plugin\Field\FieldFormatter;

use Drupal\Core\Field\FieldItemListInterface;
use Drupal\Core\Render\Element;
use Drupal\link\Plugin\Field\FieldFormatter\LinkFormatter;

/**
 * Plugin implementation of custom icon formatter.
 *
 * @FieldFormatter(
 *   id = "MY_MODULE_link",
 *   label = @Translation("Link w/ icon support"),
 *   field_types = {
 *     "link"
 *   }
 * )
 */
class LinkIconFormatter extends LinkClassFormatter {

  /**
   * {@inheritdoc}
   */
  public function viewElements(FieldItemListInterface $items, $langcode) {
    $build = parent::viewElements($items, $langcode);
    foreach (Element::children($build) as $delta) {
      if (!empty($build[$delta]['#options']['attributes']['icon'])) {
        // Modify $build[$delta] here.
      }
    }
    return $build;
  }

}