Shopping cart not cleared after checkout
This looks like a permission issue. Check the permissions page admin/people/permissions and make sure only relevant roles can edit orders. 

By default, only the following permissions should be granted to authenticated users, and to any customer role you may have created  : 


Create Order orders, optionally Create orders of any type
View own Order orders, optionally View own orders of any type


About the items remaining in shopping cart after checkout, have you tested yourself ? I'd more believe your customers edited orders from their orders page if they had permission to... Else, a custom rule or a custom module may cause that behaviour.

As explained in drupalcommerce.org


  At the heart of Drupal Commerce is the ubiquitous Order. This Order is
  first created when a user adds a product to a cart. The Order is
  actually the "cart" when it has a status of "Shopping Cart."


Cart is only one of the possible states of an order.