How can I correctly redirect user to the submitted post and define a template for each new post in WordPress?

Question

you need to get the ID from wp_insert_post and make sure you exit or die so:

$pid = wp_insert_post($new_post);
wp_redirect( get_permalink( $pid );
die();

As for the second part you can use the WordPress Template Hierarchy and simply name each one of your custom post type designs as single-{post_type}.php for example if your post type is named event then name that file single-event.php and WordPress will know to use that file for displaying all posts of event type.