Using different size thumbnails on different parts of my site... with functions.php?

The code you posted is a function definition, and function definitions belong in functions.php to begin with. Simply move the code to functions.php, and then call it from wherever you need it in your template files.

If you want a variable image sizes, you simply need to pass an argument to the function. For example:

<?php
  function mytheme_get_attached_images( $size="full" ){
    // This function runs in "the_loop", you could run this out of the loop but
    // you would need to change this to $post = $valid_post or something other than
    // using the global post declaration.
    global $post; 
    $args = array(
      'post_type' => 'attachment',
      'numberposts' => 1,
      'post_status' => null,
      'post_parent' => $post->ID,
      'order' => 'ASC',
      'orderby' => 'menu_order'
      ); 
    $attachment = get_posts($args); // Get attachment
    if ( $attachment ) {
      $img = wp_get_attachment_image_src( $attachment[0]->ID, $size ); ?>
      <img alt="<?php the_title(); ?>" src="https://wordpress.stackexchange.com/questions/26031/<?php echo $img[0] ; ?>" width="<?php echo $img[1] ?>" height="<?php echo $img[2] ?>"/>
  <?php }
  }
?>

Then, when you call the function, it would look like this:

<?php mytheme_get_attached_images( 'medium' ); ?>

(Note: there are other ways the code could be improved; this just shows you how to use the existing code for your purposes.)

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