How to echo dynamically selected categories of back-end in front end using wordpress

I have in functions.php

//Example of select field
    $this->settings['select'] = array(
        'section' => 'general',
        'title'   => __( 'Example Select' ),
        'desc'    => __( 'This is a description for the drop-down.' ),
        'type'    => 'select',
        'std'     => '',
        'choices' => array(
            'choice1' => 'Choice 1',
            'choice2' => 'Choice 2',
            'choice3' => 'Choice 3'
        )
    );

//Here i have managed to echo the categories dynamically in the back-end
$args = array(
            'orderby' => 'name',
            'order' => 'ASC',
            'hide_empty' => 0
        );
$categories = get_categories($args);
$categories_name = array();
foreach($categories as $category){
   $categories_name[] = $category->name;
}

    $this->settings['categoriesmain1'] = array(
          'section' => 'general',
          'title'   => __( 'Select Left Block Category' ),
          'desc'    => __( 'Here you can select the main category for main left block.' ),
          'type'    => 'select',
          'std'     => '',
          'choices' =>  $categories_name // this returns the category names in a select field
        );
    $settings = get_option('mytheme_options');
    $my_choices_cat1  = $settings['categoriesmain1'];

    $this->settings['categoriesmain2'] = array(
          'section' => 'general',
          'title'   => __( 'Select Center Block Category' ),
          'desc'    => __( 'Here you can select the main category for main center block.' ),
          'type'    => 'select',
          'std'     => '',
          'choices' =>  $categories_name
        );
    $settings = get_option('mytheme_options');
    $my_choices_cat2  = $settings['categoriesmain2'];

    $this->settings['categoriesmain3'] = array(
          'section' => 'general',
          'title'   => __( 'Select Right Block Category' ),
          'desc'    => __( 'Here you can select the main category for main right block.' ),
          'type'    => 'select',
          'std'     => '',
          'choices' =>  $categories_name
        );
    $settings = get_option('mytheme_options');
    $my_choices_cat3  = $settings['categoriesmain3'];

index.php

   <?php $settings = get_option('mytheme_options');
        query_posts('category_name='.$settings["categoriesmain1"].'&posts_per_page=1');
while ( have_posts() ) : the_post(); ?>
     <div class="boxes-third boxes-first">
        <div class="boxes-padding"> 
            <div class="bti">
                <div class="featured-images"><?php the_post_thumbnail(array(300,300)); ?></div>
                <div class="featured-titles"><?php echo get_the_title($post->ID); ?></div>
            </div>
            <div class="featured-text"><?php the_content('',FALSE,''); ?></div>
        </div>
        <span class="box-arrow"></span>
    </div>
    <?php endwhile; ?>
    <?php wp_reset_query(); ?>

   <?php $settings = get_option('mytheme_options');
         query_posts('category_name='.$settings["categoriesmain2"].'&posts_per_page=1');
while ( have_posts() ) : the_post(); ?>
    <div class="boxes-third">
        <div class="boxes-padding">
            <div class="bti">
                <div class="featured-images"><?php the_post_thumbnail(array(300,300)); ?></div>
                <div class="featured-titles"><?php echo get_the_title($post->ID); ?></div>
            </div>
            <div class="featured-text"><?php the_content('',FALSE,''); ?></div>
        </div>
        <span class="box-arrow"></span>
    </div>
    <?php endwhile; ?>
    <?php wp_reset_query(); ?>

   <?php $settings = get_option('mytheme_options');
         query_posts('category_name='.$settings["categoriesmain3"].'&posts_per_page=1');
while ( have_posts() ) : the_post(); ?>
    <div class="boxes-third boxes-last">
        <div class="boxes-padding">
            <div class="bti">
                <div class="featured-images"><?php the_post_thumbnail(array(300,300)); ?></div>
                <div class="featured-titles"><?php echo get_the_title($post->ID); ?></div>
            </div>
            <div class="featured-text"><?php the_content('',FALSE,''); ?></div>
        </div>
        <span class="box-arrow"></span>
    </div>
   <?php endwhile; ?>
   <?php wp_reset_query(); ?>

The problem is that in index.php the $settings[] is not echoed properly.
For example

query_posts('category_name='.$settings["categoriesmain3"].'&posts_per_page=1');,

if i add var_dump or print_r it will echo something like Array {} (an empty array) instead of echoing the category name, should be

  query_posts('category_name=Category3&posts_per_page=1'); 

where Category3 is the selected category from the back-end by the user.
My functions.txt is http://pastebin.ca/2476028
Please help me i am doing my best on learning wordpress.