How to get the post content after WordPress post processing?

If by “post processing” you mean the autoformatting, the shortcode processing, that sort of thing, then you want to pass the content through the the_content filters.

$my_id = 7;
$post_id_7 = get_post($my_id); 
$post_content = $post_id_7->post_content;
$post_content = apply_filters('the_content',$post_content);

http://codex.wordpress.org/Function_Reference/apply_filters

You can see much of what will be applied by default by looking at the source of wp-includes/default-filters.php. However, plugins and themes can add filters so you may find that you want to pick and choose what gets applieds.