How to insert category list into post creation page, and retrieve chosen categories?

By default page post type doesn’t support category taxonomy. But you can easily fix it by register category taxonomy for page post type:

add_action( 'init', 'wpse8170_init' );
function wpse8170_init() {
    register_taxonomy_for_object_type( 'category', 'page' );
}

Now you will see categories meta box in the same place as you see it at post creation page.

To get page’s categories just call wp_get_post_categories function like this (thanks to @Subharanjan):

$page_categories = wp_get_post_categories( get_the_ID() );