How to check if I’m on the last page of posts?

The WP_Query object contains a max_num_pages field which contains how many pages of posts there are. You can compare the current page number with it. (This is how get_next_posts_link() does it.)

global $wp_query;
$current_page = $wp_query->get( 'paged' );
if ( ! $current_page ) {
    $current_page = 1;
}
if ( $current_page == $wp_query->max_num_pages ) {
    // You are on the last page
}