How to export all children of a family?
The relationship report will give you all the children of a contact easily for you. Relationship reports are standard with CiviCRM and appear under Contact Reports.

Edit filters to ensure the relationship is filtering on only the type you're interested in reporting on. You can then export to csv which will allow you to sort as required there if you want them to display in a particular way.

You'll see that it can only filter on one relationship at a time, so if you put all the contacts that you want to include into a group, you can filter on that instead to display all the relationships on contact in that group.

If you are using Drupal, you might also want to explore views to display this information for you. However, if you don't use Drupal or aren't familiar with views, this is a steep learning curve! If you use it in conjuction with views data export (https://www.drupal.org/project/views_data_export) you can get the information into excel from the view also.