How to output themed results from a form submission?

Well, you're getting close.

Note that in Drupal, we don't use $_POST data, as it is unsanitized, and therefore dangerous. The sanitized form values are stored in $form_state['values'], and these are what you should use. You can then store data in $form_state['storage'], and if you set $form_state['rebuild'] to TRUE, the values you saved in $form_state['storage'] will be available in your form definition:

/**
* Implements hook_menu
*/
function myownform_menu() {
    $items = array();
    $items['myownform'] = array(
        'title' => 'My Own Form',
        'type' => MENU_NORMAL_ITEM,
        'access arguments' => array('submit myownform'),
        'page callback' => 'drupal_get_form',
        'page arguments' => array('myownform_form'),
    );
    return $items;
}
/**
* Implements hook_form
*/
function myownform_form($form, &$form_state) {
    $form['myowntext'] = array(
        '#type' => 'textfield',
        '#title' => t('My own Text'),
        '#size' => 50,
        '#maxlength' => 50,
        '#required' => TRUE,
        '#description' => t('Please enter a small text'),
    );
    $form['submit'] = array(
        '#type' => 'submit',
        '#value' => t('Submit')
    );

    // $form_state['storage'] is only set when the form
    // as been submitted (see the _submit() function for
    // more details)
    if(isset($form_state['storage'], $form_state['storage']['myowntext']))
    {
        // The form has been submitted, so we display the submitted
        // value to the user:
        $form['myowntext_display'] = array
        (
            '#markup' => t('The value you submitted was: @value', array('@value' => $form_state['storage']['myowntext'])),
            '#prefix' => '<p>',
            '#suffix' => '</p>',
        );
    }
    return $form;
}

/**
* Submission handler for the myownform_form
*/
function myownform_form_submit($form, &$form_state) {
    // First we save the submitted data to $form_state['storage']
    $form_state['storage']['myowntext'] = $form_state['values']['myowntext'];
    // Then we set $form_state['rebuild'] to TRUE, so that the values stored in
    // $form_state['storage'] are available in the form definition.
    $form_state['rebuild'] = TRUE;
}