How do you create a custom template to display a category with an image and related posts below?

I am not sure I understand exactly the problem, but If I do – there is absolutely no need for a custom post type.

all you have to do is use multiple loops and basically divide the query into sub-loops for each category .

<div id="cat-1">
     <?php query_posts('cat=1&posts_per_page=5');// put relevant category and number of posts
         if (have_posts()) : while (have_posts()) : the_post();
         // Do whatever ...
     endwhile; endif; reset_query();?>

</div>

<div id="cat-2">
     <?php query_posts('cat=2&posts_per_page=1');// put relevant category and number of posts
          if (have_posts()) : while (have_posts()) : the_post();
          // Do whatever ..
     endwhile; endif; reset_query(); ?>
</div>

<div id="cat-3">
     <?php query_posts('cat=3&posts_per_page=-1');// put relevant category and number of posts
          if (have_posts()) : while (have_posts()) : the_post();
          // Do whatever ..
     endwhile; endif; reset_query();?>
</div>..etc.. etc..

Note that you can do the same using Loops with query_posts() , Loops with WP_Query() or Loops with get_posts().

regarding the image – I am not sure what Image you want – but if you refer to a category image (special image assigned to categories) you will need to use some plugin like these ..