I would like to add a custom field for user to select a existing post or page to a post. (eg as related post, or as similar post...) a way that the user can browse trough exisisting and published posts and then select 1, via post editing screen, just like the "LINK FROM EXISTING CONTENT" tool at the TinyMCE
The excellent My Metabox class by bainternet does exactly what you are looking for. Add a post dropdown list with
$my_meta->addPosts('posts_field_id',array('post_type' => 'post'),array('name'=> 'My Posts '));
And it will create a meta box with posts dropdown (you can change this to a checkbox list if you want to be able to select more than one post)
My Metabox class can be customized to your liking with little effort.
Edit: It's My Metabox Class not Tax Meta Class, which lets you create taxonomy meta data in the same way.
http://www.advancedcustomfields.com/ is a great plugin that allows you to add all sorts of custom fields in your post/page screens.
One of the fields it lets you add is a "Post Object" field type. This lets you select any post in your blog and then display some/all of the information from that post. See the documentation on the plugin here: http://www.advancedcustomfields.com/resources/field-types/post-object/