<div class=carousel>
<ul class=panes>
<?php
// the query
$args = array('cat' => id_of_featured_posts_category, 'posts_per_page' => 5);
$the_query = new WP_Query( $args ); ?>
<?php if ( $the_query->have_posts() ) : ?>
<?php while ( $the_query->have_posts() ) : $the_query->the_post(); ?>
<li>
<h2><a href="https://wordpress.stackexchange.com/questions/157282/<?php the_permalink(); ?>"><?php the_title(); ?></a></h2>
<?php the_post_thumbnail( 'full'); ?>
</li>
<?php endwhile; ?>
<?php endif; ?>
</ul>
</div>
I don’t have wordpress here to test it, but it should work, to learn more about this you can check the documentation for WP_Query. I don’t know where you want to add the excerpt (the_excerpt() is the function you need to call to get it), but I’m sure you can take from here.