Posts from multiblt categories but featured image only in the first one, the rest to have the post title only

General notes

1. 1^st of all: You don’t have to use the (imho crappy) TimThumb thingy. There’re much better things like my (freely available, open source) »Dynamic image resize« plugin.
2. 2^nd (and afaik), you can do this with on board stuff, like get_image_tag() in your case.
3. 3^rd: Don’t throw that many variables into the global namespace. If you really think you can’t work around it: prefix them with a unique string like wpse69602_.
4. 4^th Try to write clean code with lots of white space. This helps you spotting errors like this: 'include' => '4,5,6,7'. Hint: Most of those functions take arrays and not special-character separated strings.

Your cleaned up code

The kool thing I used to identify the number of the currently looped post isn’t a $flag. It’s a part of the $wp_query object: The current_post, which is a flag already set by WordPress.

<div class="container">
<?php
$terms = get_terms(
     'category'
    ,array(
         'include' => array( '4,5,6,7' )
        ,'orderby' => 'name'
        ,'order'   => 'ASC'
    )
);
if ( ! empty( $terms ) )
{
    foreach( $terms as $term ) 
    {
        $my_query = new WP_Query( array(
             'category__in'     => array( $term->term_id )
            ,'post_type'        => 'post'
            ,'post_status'      => 'publish'
            ,'posts_per_page'   => 1
            ,'caller_get_posts' => 1
        ) );
        if ( $my_query->have_posts() )
        {
            while ( $my_query->have_posts() )
            {
                $my_query->the_post();
                ?>

                <div class="post_home_main">

                <!-- Show the posts featured image -->
                <?php
                if ( 1 <= $my_query->current_post )
                {
                    if ( $image_url = wp_get_attachment_url( get_post_thumbnail_id() ) )
                    {
                        echo get_image_tag(
                             get_post_thumbnail_id()
                            ,__( 'Featured Image', 'your_textdomain' )
                            ,get_the_title()
                            ,'none'
                             // Check your image sizes and adjust this one
                            ,'thumb'
                        );
                    }
                    else
                    {
                        ?>
                        <img src="https://wordpress.stackexchange.com/questions/69602/<?php echo get_template_directory()."/images/no-image.jpg'; ?>" alt="No image available" class="panikosule" />
                        <?php
                    }
                }
                ?>

                <a href="<?php the_permalink() ?>" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a>
                <?php the_excerpt(); ?> 

                </div>

                <?php
            } // endwhile;
        }
    }
}
wp_reset_query();  // Restore global post data stomped by the_post().
?>
<span class="arrow"></span>
</div>