Category template not displaying all post formats

Reading the codex on get_template_part and get_post_format will help you a lot here.

It’s hard to say for sure without knowing what files are in your theme but get_template_part( 'content', get_post_format() ); is essentially saying use the template named content-format.php, where format is one of image, video, gallery etc. One of a few things is happening:

Either your theme doesn’t contain these templates at all, although you should still see posts in the fallback single.php format if that were the case. Still worth checking if they exist as no point using get_post_format()if they don’t.

Or those templates are in a sub directory down form your category template in which case you’d need the format get_template_part( 'subfolder/content', get_post_format() );

Or your posts all have the default/standard format in which case get_post_format() will return false. If this is the case then you could remove get_post_format() and use just get_template_part( 'content' ); where all posts will use the default content template part. Alternatively you can assign a value to the false response:

get_header();
if ( have_posts() ) :
    while ( have_posts() ) : the_post();
    $format = get_post_format();
    if ( false === $format ) {
    $format="standard";
                            }
        get_template_part( 'content', $format );
    endwhile;
    get_template_part( 'includes/navigation', 'index' );
else:
    get_template_part( 'includes/no-results', 'index' );
endif;
get_footer();

Here you’d need to insure you added a content-standard.php alongside the other format template parts.