Custom header image, and content, on Posts Page in twentyeleven with Static Front Page

if you work with a static front page, the ID for the posts page is get_option('page_for_posts').

try and replace this section:

            <?php
            /*
             * The header image.
             * Check if this is a post or page, if it has a thumbnail, and if it's a big one
             */
            if ( is_singular() && has_post_thumbnail( $post->ID ) &&
                    ( /* $src, $width, $height */ $image = wp_get_attachment_image_src( get_post_thumbnail_id( $post->ID ), array( $header_image_width, $header_image_width ) ) ) &&
                    $image[1] >= $header_image_width ) :
                // Houston, we have a new header image!
                echo get_the_post_thumbnail( $post->ID, 'post-thumbnail' );

with something like:

            <?php //edited to show the selected featured image if a static page is selected as posts page
            // The header image
            // Check if this is a post or page, if it has a thumbnail, and if it's a big one
            if( is_singular() ) $post_id = $post->ID;
            elseif ( is_home() && get_option('page_for_posts') ) $post_id = get_option('page_for_posts');
            if ( ( is_singular() || is_home() && get_option('page_for_posts') ) &&
                    has_post_thumbnail( $post_id ) &&
                    ( /* $src, $width, $height */ $image = wp_get_attachment_image_src( get_post_thumbnail_id( $post_id ), array( $header_image_width, $header_image_width ) ) ) &&
                    $image[1] >= $header_image_width ) :
                // Houston, we have a new header image!
                echo get_the_post_thumbnail( $post_id, 'post-thumbnail' );

(not tested in wp3.7.1)

but any content I have on the recent-news page isn’t shown; its only displayed content is the summaries of the Posts

that is the way the posts page works when set together with a static front page, as it is using the index template – http://codex.wordpress.org/Creating_a_Static_Front_Page