These are completely different requests.
It looks like you are using includes, which has nothing to do with querying the database for the content (posts and metadata) of a category or categories...
To get 6 posts from different categories, you'd have to use a get_post() function
and build an array to output the content as you want it displayed. However, I'd go with the custom query I mentioned above and use different categories for the different 6 things you want to display... particularly if you are trying to display the most recent post in each. The custom query would look like:
<?php $my_query = new WP_Query('cat=1,2,3,4,5,6&showposts=1'); ?>
<?php while ($my_query->have_posts()) : $my_query->the_post(); ?>
<h2><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></h2>
<?php the_excerpt(); ?>
<p >Read the rest of "<?php the_title(); ?>" in <?php the_category(' » '); ?></p>
<p class="postmetadata"><?php edit_post_link('Admin Edit', '', ''); ?></p>
<?php endwhile; ?>
That would dislay the latest post from each category (1, 2, 3, 4, 5, 6) which *isn't* the same thing as the last 6 posts from ONE category.
You have to think about how you structure your content before you design the code to get the content. You *could* make the 6 categories as *sub-categories* of that ONE category... I can't take time to figure out how to merge your intention with your content - but once you have it actually figured out logically, then decide if you wnat 6 posts from 1 cat or 1 post from 6 cats...