Have posts included on a page that share same category

(…) are pages in fact posts?

You won’t think what every thing is a post type in WordPress: Posts, Pages, Links, Nav Menu items… The same goes for Terms, Categories and other things, which are built in Taxonomies.

You can take a look at it with adding the following very late in your theme (the footer for example).

echo '<pre>'.var_export( wp_list_pluck( $GLOBALS['wp_post_types'], '_builtin' ), true ).'</pre>';

(…) “static” or evergreen page content at the top, and posts sharing the same category underneath (…)

You’re searching for »Sticky« posts.