how to randomly list 5 authors with at least 3 published posts

You could use count_many_users_posts().

$args = array(
    'exclude' => array( 4, ),
    'fields' => 'ID',
);
$users = get_users( $args );
$user_posts = count_many_users_posts( $users );
foreach( $user_posts as $user => $posts ) {
    if( $posts < 3 ) {
        unset( $user_posts[$user] );
    }
}
$user_ids = array_keys( $user_posts );
shuffle( $user_ids );
for( $i = 0; $i <=4; $i++ ) {
    $user = get_userdata( $user_ids[$i] );
    // display your user info here
}

Reference

• count_many_users_posts()
• get_users()
• array_keys()