I want Page titles and excerpts to show up on home.php in certain order

Unless you have a particular reason to have meta box order values , that would be overkill.

Instead ditch the excerpt for pages, and add a meta box called “intro_content” (or whatever).

Then in a home.php query you can easily grab this info and order it however you want using html.

For instance:

// replace 1, 2, 3, with actual page ID's
$about_meta = get_post_meta( 1, 'intro_content');
$new_patients_meta = get_post_meta( 2, 'intro_content'); 
$service_meta = get_post_meta( 3, 'intro_content'); 

// div styles and echo the title + the above or whatever

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

Added this after your comment/edit.

To order the metabox you can add another one with a numerical value, or even better add a dropdown box ( for example called ‘order_value_dropdown’).

That way in your query you can query the order by value of the dropdown. For example :

 $order_meta = get_post_meta( 3, 'order_value_dropdown'); 
 if ($order_meta = 1 )
 //grab the intro_content meta from that post
 if ($order_meta = 2 )
 //grab the intro_content meta from that post
 // etc, etc