Display custom post types in wp_list_pages

To simplify things, first make your archives actual Archives – not Pages. That will help you later on in listing the correct content.

unregister_post_type('blog');
unregister_post_type('event');
unregister_post_type('news');

Unregistering the post type won’t delete any content, it will just ensure that when you re-register them, the settings will take effect.

Tweak your register_post_type calls which should look something like this now:

register_post_type('blog',
    array(
        ... a bunch of settings ...
    )
);

and adjust, or add, has_archive and rewrite. Change the “news-and-events” part to whatever your News and Events Page slug is:

register_post_type('blog',
    array(
        ... a bunch of settings ...,
        'has_archive' => 'news-and-events/blog',
        'rewrite' => array('slug' => 'news-and-events/blog'
    )
);

(do the same for all 3 post types, swapping out “blog” for the current post type)

At this point, you’ll need to delete the Pages, so WordPress doesn’t get confused about whether to display a Page or an Archive. You should also visit the Settings > Permalinks page in wp-admin. You don’t need to change any settings; just visiting this page will force WP to update permalinks and ensure you can access the new Archives.

Then, create or edit archive.php in your theme. (If you’re not already using a child theme or a custom theme, stop and create a child theme, so that when the parent theme updates, you won’t lose your edits.) You can now add a conditional so that whenever you are on a Blog, Event, or News archive, you see that sidebar.

// Check the main query to see if this is one of the archives we're targeting.
$current_post_type = $wp_query->query['post_type'];
// If so:
if($current_post_type == 'blog' || $current_post_type == 'event' || $current_post_type == 'news') {
    // Display the sidebar. (edit 'name-of-your-sidebar')
    if(!function_exists('dynamic_sidebar') || !dynamic_sidebar('name-of-your-sidebar')): endif;
}
// Otherwise, do nothing here. Or, insert a different sidebar that will apply to all other archives, such as Post Categories.

It sounds like you don’t have the sidebar set up as a dynamic widgetized sidebar. So, you could either replace the “dynamic_sidebar” call above with a hard-coded sidebar, or turn your post listing code into a widget. I would recommend that approach, as you can then more easily control where it appears on the site.