Alphabetical search

Yes. Do a SQL query for posts using LIKE 'x%', where x is your letter.

Here’s a simple example:

global $wpdb, $post;

$letter="a"; // or $_GET['letter']...

$query = "SELECT * FROM {$wpdb->posts} WHERE post_name LIKE %s";
$query = $wpdb->prepare($query, $letter.'%');

$results = $wpdb->get_results($query);

// note that variable name must be $post, 
// because you need to override stupid wp globals...    
foreach($results as $post){

  setup_postdata($post);

  // the usual loop functions go here
  the_title();

}

wp_reset_postdata();