How can I make this process automatic?

I agree with @brasofilo that this is two very different questions, but I think we can answer both here.

1. Automatically add child pages of A to custom taxonomy B’s term C.

We’ll add the term ‘yummy’ in the custom taxonomy ‘food-adjective’ to all child pages of page ID 123 when they’re published. You can add this to your theme’s functions.php file.

/**
 * Automatically add a custom taxonomy term to child pages of page 123 on publish
 *
 * @param int $post_id 
 * @param object $post The new post
 * @return void
 */
function wpse_74605_auto_tax( $post_id, $post ) {
    if ( 123 === $post->post_parent ) {
        wp_set_post_terms( $post_id, 'yummy', 'food-adjective', true );
    }
}
add_action( 'publish_page', 'wpse_74605_auto_tax', 10, 2 );

2. Automatically add child pages to menu

This is covered in a number of plugins, so I won’t reinvent the wheel here. I’d recommend Viper007bond’s Add Descendants as Submenu Items.