Style adjustment

Hi Luca

Consider creating a child theme instead of editing a theme directly — if you upgrade the theme all your modifications will be lost.

To add the excerpt to the caption put this in your functions.php file:

add_filter( 'related_posts_by_taxonomy_caption', 'rpbt_add_excerpt_to_caption', 10, 2 );

function rpbt_add_excerpt_to_caption( $caption, $related ){
	$caption =  trim( $caption );

	$caption = $caption ? '<h4>' . $caption . '</h4>' : '';

	global $post;
	$post = $related;
	setup_postdata( $related );

	// get's the autogenerated excerpt or post execerpt if available
	$excerpt = apply_filters( 'the_excerpt', get_the_excerpt() );
	wp_reset_postdata();

    $caption .= $excerpt ? $excerpt : '';
    return $caption;
}

To style the gallery see the selectors you can use here:
https://keesiemeijer.wordpress.com/related-posts-by-taxonomy/post-thumbnails/#css-selectors

To style the post title in the caption

.related-gallery .gallery-caption h4 {

}

To style the excerpt in the caption

.related-gallery .gallery-caption p {

}

I assume by site title you mean the widget or shortcode title.

The widget title can be styled with this selector

.widget.related_posts_by_taxonomy .widget-title {

}

The shortcode title with this:

.rpbt_shortcode h3 {

}