Can a household register online?
If you are using Drupal as your CMS, then the answer is "yes". The drupal Webform-CiviCRM Integration module allows you to set up a form to register one or more people for any number of events, with full control over creating relationships, sharing addresses, sending a concise email recipt, and customizing the look and feel of your form.

For more information, see the user manual, or this how-to video from CiviTeacher.