Retrieving featured image on static blog posts page

In your example, $posts_page contains the ID of the page for posts, you can use that with any API function that accepts a post ID.

For the featured image:

$posts_page = get_option( 'page_for_posts' );
$post_thumbnail_id = get_post_thumbnail_id( $posts_page );
$url = wp_get_attachment_image_src( $post_thumbnail_id, 'your-custom-size' );
echo $url;

And outputting post title with filters applied:

$posts_page = get_option( 'page_for_posts' );
$title = get_post_field( 'post_title', $posts_page );
if ( ! is_wp_error( $title ) ) {
    echo $title;
}