Contact form with ajax + save form in bdd

Hello,
I have a probleme with a dev from one day so i post here. Maybe someoine have a solution.
I created à custom contact form in a template wordpress. Form is basic (name, surname…).
The form is in the template and send data via ajax to a process.php files which is in my template folder.
The process.php files receive and answer the data but don’t save in BDD the DATA. But if i call the process.php file manually it’s ok.
SO i think it’s AJAX which don’t do what i want. But i don’t see how to do that.

My form is like that :

v id="form_contact">
    <form action="<?php echo get_bloginfo('stylesheet_directory')?>/process.php" id="contact" method="POST">
	
	<div id="msgok" style="display:none" >
	<div class="alert alert-danger"><button class="close" type="button" data-dismiss="alert" aria-hidden="true">×</button><strong>Message envoyé</strong></div>
	</div>
        <p>
            <label for="nom" class="nom">Nom</label>
            <br /><input id="nom" name="nom" type="text">
            <span id="msg_nom"></span>
        </p>
        <p>
            <label for="sujet" class="sujet">Sujet</label>
            <br /><input id="sujet" name="sujet" type="text">
            <span id="msg_sujet"></span>
        </p>
        <p>
            <label for="email">Email</label>
            <br /><input id="email" name="email" type="email">
            <span id="msg_email"></span>
        </p>
        <p>
            <label for="message">Message</label>
            <br /><textarea id="message" name="message" rows="10" cols="80"></textarea>
           <span id="msg_message"></span>
        </p>
        <p>
            <input type="submit" value="Envoyer" />
        </p>
    </form>
    <span id="msg_all"></span> 
</div><!-- end of #form_contact -->

My php file like that

<?php
   
 ////test save bdd ////
   
      if ($_SERVER["REQUEST_METHOD"] == "POST") {
 require_once($_SERVER['DOCUMENT_ROOT'].'/site3/wp-load.php');
	require_once ($_SERVER['DOCUMENT_ROOT'].'/site3/wp-config.php');
		global $wpdb;
$table_name = $wpdb->prefix . "contact_perso";
$wpdb->insert($table_name , array('id' => "" ,'nom' => "123" ,'sujet' => "456",'email' => "0", 'message' => "123", 'id_auteur' => "123") );
	  }
  
		
		////test save bdd ////  
   
   
   
    // CONDITIONS NOM
    if ( (isset($_POST["nom"])) && (strlen(trim($_POST["nom"])) > 0) ) {
        $nom = stripslashes(strip_tags($_POST["nom"]));
    } else {
        echo "Merci d'écrire un nom <br />";
        $nom = "";
    }

    // CONDITIONS SUJET
    if ( (isset($_POST["sujet"])) && (strlen(trim($_POST["sujet"])) > 0) ) {
        $sujet = stripslashes(strip_tags($_POST["sujet"]));
    } else {
        echo "Merci d'écrire un sujet <br />";
        $sujet = "";
    }

    // CONDITIONS EMAIL
    if ( (isset($_POST["email"])) && (strlen(trim($_POST["email"])) > 0) && (filter_var($_POST["email"], FILTER_VALIDATE_EMAIL)) ) {
        $email = stripslashes(strip_tags($_POST["email"]));
    } elseif (empty($_POST["email"])) {
        echo "Merci d'écrire une adresse email <br />";
        $email = "";
    } else {
        echo "Email invalide :(<br />";
        $email = "";
    }

    // CONDITIONS MESSAGE
    if ( (isset($_POST["message"])) && (strlen(trim($_POST["message"])) > 0) ) {
        $message = stripslashes(strip_tags($_POST["message"]));
    } else {
        echo "Merci d'écrire un message<br />";
        $message = "";
    }

    // Les messages d'erreurs ci-dessus s'afficheront si Javascript est désactivé

    // PREPARATION DES DONNEES
    $ip           = $_SERVER["REMOTE_ADDR"];
    $hostname     = gethostbyaddr($_SERVER["REMOTE_ADDR"]);
    $destinataire = "[email protected]";
    $objet        = "[Site Web] " . $sujet;
    $contenu      = "Nom de l'expéditeur : " . $nom . "\r\n";
    $contenu     .= $message . "\r\n\n";
    $contenu     .= "Adresse IP de l'expéditeur : " . $ip . "\r\n";
    $contenu     .= "DLSAM : " . $hostname;

    $headers  = "CC: " . $email . " \r\n"; // ici l'expediteur du mail
    $headers .= "Content-Type: text/plain; charset=\"ISO-8859-1\"; DelSp=\"Yes\"; format=flowed /r/n";
    $headers .= "Content-Disposition: inline \r\n";
    $headers .= "Content-Transfer-Encoding: 7bit \r\n";
    $headers .= "MIME-Version: 1.0";

    // SI LES CHAMPS SONT MAL REMPLIS
    if ( (empty($nom)) && (empty($sujet)) && (empty($email)) && (!filter_var($email, FILTER_VALIDATE_EMAIL)) && (empty($message)) ) {
        echo 'echec :( <br /><a href="contact.html">Retour au formulaire</a>';
    } else {
        // ENCAPSULATION DES DONNEES ok
		
		
        mail($destinataire, $objet, utf8_decode($contenu), $headers);
        echo 'Formulaire envoyé';
    }

    // Les messages d'erreurs ci-dessus s'afficheront si Javascript est désactivé
?>

and JS script as :

<script>
    $(function(){
        $("#contact").submit(function(event){
            var nom        = $("#nom").val();
            var sujet      = $("#sujet").val();
            var email      = $("#email").val();
            var message    = $("#message").val();
            var dataString = nom + sujet + email + message;
            var msg_all    = "Merci de remplir tous les champs";
            var msg_alert  = "Merci de remplir ce champs";

            if (dataString  == "") {
                $("#msg_all").html(msg_all);
            } else if (nom == "") {
                $("#msg_nom").html(msg_alert);
            } else if (sujet == "") {
                $("#msg_sujet").html(msg_alert);
            } else if (email == "") {
                $("#msg_email").html(msg_alert);
            } else if (message == "") {
                $("#msg_message").html(msg_alert);
            } else {
                $.ajax({
                    type : "POST",
                    url: $(this).attr("action"),
                    data: $(this).serialize(),
                    success : function() {
					document.getElementById("msgok").style.display = "";
						setTimeout(function() {
							document.getElementById("contact").style.display = "none";
						}, 3000);
						
                    },
                    error: function() {
                        $("#contact").html("<p>Erreur d'appel, le formulaire ne peut pas fonctionner</p>");
                    }
                });
            }

            return false;
        });
    });
</script>