Display all post types together

For the first part of your question, a pre_get_posts hook is all you would need. This code should work, providing you are using the default “Latest Posts” setting as your front page (Settings > Reading).

add_action( 'pre_get_posts', 'se132019_home_page_all_post_types' );
function se132019_home_page_all_post_types( $query ) {
    if ( ! is_admin() $query->is_main_query() && $query->is_posts_page() ) {
        $query->set( 'post_type', 'any' );
    }
}

If you are using a custom WP_Query, just add 'post_type' => 'any' as an argument.

The second piece is a personal preference for you – you could just house all posts in Posts, and display them differently depending on the terms (categories) they are attached to. Otherwise, you would need to do the following 2 things:

1. Add your custom post types (Job Listings) to the category taxonomy. This can be done when registering the post type using taxonomies argument, or using register_taxonomy_for_object_type.
2. hook into the save_post method, check to see if the post was attached to any Categories. If not, manually assign it to the proper category, depending on which post type it was.

EDIT::

your first couple lines should look like the following. When a WP_Query object is created, it automatically calls the query() function. You are calling it a second time, passing it new arguments.

        <?php 
        $args = array(
            'post_type' => 'any',
            'posts_per_page' => 3
        );
        $recent = new WP_Query($args);