counting trackbacks, pingbacks and comments of a post

I think you can use get_comments.

In the type parameter you can pass the type of comment (trackback, ping), you can filter by status, and you have a parameter called count, that when is set to true returns just the number of comments that match the parameters you passed.

 get_comments(
    array(
    'status' => 'approve',
    'post_id'=> get_the_ID(), 
    'type'=> 'pings', 
    'count' => true)
 );

This example gets the number of approved pings for a post, you have to use it inside a loop, if not, the function get_the_ID() wont work, but you just have to pass the ID as you want.