Well, pooh! After two evening of work, I finally even got the children documents fixed.

And learned something in the process! You can use a specific value in a "SELECT" query! So it didn’t need any UPDATE queries after the document was created after all!
$sql = "SELECT type, contentType, pagetitle, longtitle, description, published, pub_date, unpub_date, '$parent' as 'parent', isfolder, introtext, content, richtext, template, menuindex, searchable, cacheable, ".$_SESSION['mgrInternalKey'].", ".time().", editedby, editedon, deleted, deletedon, deletedby, menutitle, donthit, privateweb, privatemgr, content_dispo, hidemenu FROM $dbase.".$table_prefix."site_content WHERE id IN ($docs_to_duplicated);";
So the time wasn’t really wasted since I learned something that’s very nice to know.