How do I alter the behavior of the form fields contained within the paragraph field?
Since paragraphs^1.3 you can use hook_field_widget_WIDGET_TYPE_form_alter() (I linked the 8.2 version because there is a good comment about using the hook). 

You can see the following example in modules/contrib/paragraphs/tests/modules/paragraphs_test/paragraphs_test.module

/**  
 * Implements hook_field_widget_WIDGET_TYPE_form_alter().  
 */  
function paragraphs_test_field_widget_entity_reference_paragraphs_form_alter(&amp;$element, &amp;$form_state, $context) {
  if ($element['#paragraph_type'] == 'altered_paragraph') {
    $element['subform']['field_text']['widget'][0]['#title'] = 'Altered title';
  }
}