Q1. When we add new posts, a frequent occurrence, is the Search & Filter cache updated automatically to include them?
A1. Yes, as long as they are added via the WP editor page. If they are added though import, we have an action that the import process needs to call to add the new post (or changed post) to our cache:

https://searchandfilter.com/documentation/action-filter-reference/#update-cache-for-a-particular-post

If you are using WP All Import, then this code (in the child theme functions.php file) should be used:

add_action('pmxi_saved_post', 'wp_all_import_post_saved', 10, 1);

function wp_all_import_post_saved($id) {
    do_action('search_filter_update_post_cache', $id);
}

Q2. Is it possible to have [the form] a header section, and arranged horizontally?
A2. It depends how your theme allows you to place widgets. You may need to use a child theme and edit the template PHP file and place it like so. This page in the documentation covers this and the horizontal appearance:

https://searchandfilter.com/documentation/getting-started/display-search-form/