Custom Post Type has wrong label and is not found when called by a loop

Question

The issue with the labels in problem 1 is that you haven’t passed the labels to register_post_type(). You’ve defined the labels in the $labels variable, but this is what you passed to register_post_type():

$args = array(
    'labels' => 'Bateau',
    // ...
);

register_post_type( 'bateau', $args );

You need to pass the $labels array:

$args = array(
    'labels' => $labels,
    // ...
);

register_post_type( 'bateau', $args );

For problem 2, your problem is that you’re trying to pass a WP_Query instance to another WP_Query. $query needs to just be the array of arguments, but you’ve passed an entire WP_Query object:

<?php
    $query = new WP_Query([
        'post_type' => 'bateau',
        'posts_per_page'=>15
    ]);
?>
<?php
    $the_query = new WP_Query();
    $the_query->query($query);
; ?>

That needs to be just:

<?php
$the_query = new WP_Query ( [
    'post_type'      => 'bateau',
    'posts_per_page' =>15
] );
?>

But, you shouldn’t need to create a page or template like this at all. In your post type registration you have:

'has_archive' => true,

This means that an archive (list) page for your post type is created automatically at http://example.com/bateau/. This is what the “Voir les articles” link in the admin bar in your screenshot is linked to. If this isn’t working, make sure to go to Settings > Permalinks to flush your permalinks (visiting the settings page is enough), and make sure that your theme’s templates are properly using the main loop:

if ( have_posts() ) : 
    while ( have_posts() ) : the_post();

    endwhile;
endif;

Note the lack of $the_query-> and any new WP_Query().