I use the following code to count user posts by specific post type:

function count_user_posts_by_type( $userid, $post_type ) {
    global $wpdb;
    $where = get_posts_by_author_sql( $post_type, TRUE, $userid );
    $count = $wpdb->get_var( "SELECT COUNT(*) FROM $wpdb->posts $where" );
    return apply_filters( 'get_usernumposts', $count, $userid );
}

// Echo Count
echo count_user_posts_by_type( $user_ID, 'reports' );

Looking for an elegant solution to limit posts by dates. For example, last week\month posts.

I use the following code to count user posts by specific post type:

function count_user_posts_by_type( $userid, $post_type ) {
    global $wpdb;
    $where = get_posts_by_author_sql( $post_type, TRUE, $userid );
    $count = $wpdb->get_var( "SELECT COUNT(*) FROM $wpdb->posts $where" );
    return apply_filters( 'get_usernumposts', $count, $userid );
}

// Echo Count
echo count_user_posts_by_type( $user_ID, 'reports' );

Looking for an elegant solution to limit posts by dates. For example, last week\month posts.