Wordpress - Programmatically publish a post (custom post type) with custom fields

Use wp_insert_post() and add_post_meta(), like this:

// insert the post and set the category
$post_id = wp_insert_post(array (
    'post_type' => 'your_post_type',
    'post_title' => $your_title,
    'post_content' => $your_content,
    'post_status' => 'publish',
    'comment_status' => 'closed',   // if you prefer
    'ping_status' => 'closed',      // if you prefer
));

if ($post_id) {
    // insert post meta
    add_post_meta($post_id, '_your_custom_1', $custom1);
    add_post_meta($post_id, '_your_custom_2', $custom2);
    add_post_meta($post_id, '_your_custom_3', $custom3);
}

In addition to the great answer of @webaware above, this can be handled since wordpress 4.4.0 all via the wp_insert_post call:

$post_id = wp_insert_post(array (
    'post_content' => $content,
    'post_title' => $title,
    'post_type' => 'your_custom_post_type',
    'post_status' => 'publish',

    // some simple key / value array
    'meta_input' => array(
        'your_custom_key1' => 'your_custom_value1',
        'your_custom_key2' => 'your_custom_value2'
        // and so on ;)
    )
));

if ($post_id) {
    // it worked :)
}

This can be achieved quite easily using the Gravity Forms plugin. You can build a form which populates a Custom Post Type in the backend. This post can be set to appear as a draft or as published. No problem adding custom fields. In my case, I used it to gather client testimonials.