Media (images, videos, etc.) not showing on Category page

Your entry.php template is completely wrong. This should be your loop so to speak. Yet this is more like a page template, that also calls for entry.php. This template should include the call to your the_content(), media files, comments if applicable, and the title, also all other things you might want to include in your loop

You should have something like this in your entry.php. This is from content.php from the twentyfourteen theme

<article id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
    <?php twentyfourteen_post_thumbnail(); ?>

    <header class="entry-header">
        <?php if ( in_array( 'category', get_object_taxonomies( get_post_type() ) ) && twentyfourteen_categorized_blog() ) : ?>
        <div class="entry-meta">
            <span class="cat-links"><?php echo get_the_category_list( _x( ', ', 'Used between list items, there is a space after the comma.', 'twentyfourteen' ) ); ?></span>
        </div>
        <?php
            endif;

            if ( is_single() ) :
                the_title( '<h1 class="entry-title">', '</h1>' );
            else :
                the_title( '<h1 class="entry-title"><a href="' . esc_url( get_permalink() ) . '" rel="bookmark">', '</a></h1>' );
            endif;
        ?>

        <div class="entry-meta">
            <?php
                if ( 'post' == get_post_type() )
                    twentyfourteen_posted_on();

                if ( ! post_password_required() && ( comments_open() || get_comments_number() ) ) :
            ?>
            <span class="comments-link"><?php comments_popup_link( __( 'Leave a comment', 'twentyfourteen' ), __( '1 Comment', 'twentyfourteen' ), __( '% Comments', 'twentyfourteen' ) ); ?></span>
            <?php
                endif;

                edit_post_link( __( 'Edit', 'twentyfourteen' ), '<span class="edit-link">', '</span>' );
            ?>
        </div><!-- .entry-meta -->
    </header><!-- .entry-header -->

    <?php if ( is_search() ) : ?>
    <div class="entry-summary">
        <?php the_excerpt(); ?>
    </div><!-- .entry-summary -->
    <?php else : ?>
    <div class="entry-content">
        <?php
            the_content( __( 'Continue reading <span class="meta-nav">&rarr;</span>', 'twentyfourteen' ) );
            wp_link_pages( array(
                'before'      => '<div class="page-links"><span class="page-links-title">' . __( 'Pages:', 'twentyfourteen' ) . '</span>',
                'after'       => '</div>',
                'link_before' => '<span>',
                'link_after'  => '</span>',
            ) );
        ?>
    </div><!-- .entry-content -->
    <?php endif; ?>

    <?php the_tags( '<footer class="entry-meta"><span class="tag-links">', '', '</span></footer>' ); ?>
</article><!-- #post-## -->

It is a good idea to have a look at the default themes shipped with wordpress.