Accessing user input
PHP isn't Java...there's no concept of 'instance variables' in this context, and no automatic persistence of such vars across page requests. 

Drupal Commerce persists information about the order, order lines, products, etc to the database during the checkout process, and then re-loads the relevant entity when it needs it (which is the standard workflow).

In hook_commerce_checkout_complete() you already have access to the $order entity, so there's no need to load it yourself. I'd advise using an entity_metadata_wrapper() to access the order; to get the order lines, for example, you'd use something like:

function MYMODULE_commerce_checkout_complete($order) {
  $order_wrapper = entity_metadata_wrapper('commerce_order', $order);
  foreach ($order_wrapper-&gt;commerce_line_items as $line_item_wrapper) {
    // Do something with $line_item_wrapper.
    $field_val = $line_item_wrapper-&gt;field_something-&gt;value();
  }
}