How do I create a php page that has access to the Drupal API?
The correct way to "create a PHP page that executes Drupal API functions" is to create a module.

In that module, you implement hook_menu() which allows you to tell Drupal that it should call a given php function for a specific path, for example 'yourmodule/somepath'. Inside that, you can then execute whatever PHP and Drupal API functions which you want.

See http://drupal.org/node/1074360 for more information.