Hiding the keyword seach

I'm not sure to get the question right, but you could:

• Hide the Keyword search block by disabling the block. To do so, uncheck "Provide Keyword search block" in your faceted search search environment's settings.
• Show the Keyword search block under the Guided search block by adjusting the block's weight, in Administer > Site building > Blocks.

Aaah, I see! You could reorder (or remove) these through hook_form_alter() in a small module. Unfortunately, at the moment no other way has been designed for doing this.

More easily, you could completely hide the keyword search fieldset with CSS.

Theres a very easy way to do this as this area is helpfully themeable

stick this in your template.php - or create one in your theme folder if you don't have one yet. I'm using zen, but if you're not replace the word zen with the name of your theme.

function zen_faceted_search_ui_stage_select($search, $keyword_search_content, $guided_search_content) {
  if ($keyword_search_content) {
    $form['keyword'] = array(
    
    );
  }
  if ($guided_search_content) {
    $form['guided'] = array(
      '#type' => 'fieldset',
      '#title' => t('Guided search'),
      '#value' => $guided_search_content,
      '#weight' => 1,
      '#attributes' => array('class' => 'faceted-search-guided'),
    );
  }
  return drupal_render($form);
}

artfo's solution is great! You could even remove the whole if ($keyword_search_content) { } block.

Almost! You'd have to copy the original theme_faceted_search_ui_stage_select() function from faceted_search_ui.module into your theme's template.php, replace the "theme" word with your theme's name, and then just swap the weight values on the '#weight' => lines.