List posts from a category and embed them into a page

That’s certainly possible. Use WP_Query to search for posts, then output them as you please.

For example, to get 20 posts from the category with the id 4:

<?php
$query = new WP_Query(
    array(
        'category__in' => 4
        'posts_per_page' => 20
    )
);

if ( $query->have_posts() ) {
    echo "Here are some posts from that category:";
    echo '<ul>';
    while ( $query->have_posts() ) {
        $query->the_post();
        echo '<li><a href="' . get_permalink() . '">' . get_the_title() . '</a></li>';
    }
    echo '</ul>';
    wp_reset_postdata();
}

To query other post types or taxonomies, look at the documentation, it has lots of examples.