Resolved

To remove the Class I added this snipped to my function.php

function remove_image_class( $classes ) {
  $unneededClasses = array( 'size-medium' );

  foreach ($unneededClasses as $unneededClass) {
    $classes = str_replace($unneededClass . ' ', '', $classes);
  }

  return $classes;
}

add_filter( 'get_image_tag_class', 'remove_image_class' );

As you see, I did define an array which contains all the unneeded classes (only one in my case) so it’s easier to add more classes if needed.

To mate the paths relative I added this snippet to the functions.phpof my theme:

function relative_path_for_images($html, $id, $caption, $title, $align, $url, $size, $alt) {
  $imageurl = wp_get_attachment_image_src($id, $size);
  $relativeurl = wp_make_link_relative($imageurl[0]);
  $html = str_replace($imageurl[0],$relativeurl,$html);

  return $html;
}

add_filter('image_send_to_editor', 'relative_path_for_images', 10, 8);