How to show ‘login error’ and ‘lost password’ on my template page?

Yes it’s possible. You have to create three filters for this.

Admin stuff inside a class

class WPSE29338_Admin {
    public static function setup() {
        add_filter('login_url', array(__CLASS__, 'modifyLoginURL'), 10, 2);
        add_filter('lostpassword_url', array(__CLASS__, 'modifyLostPasswordURL'), 10, 2);
        add_filter('register', array(__CLASS__, 'modifyRegisterURL'));
    }

    public static function modifyLoginURL($loginUrl, $redirect="") {
        $loginUrl = site_url('login'); // Link to login URL

        if(!empty($redirect)) {
            $loginUrl = add_query_arg('redirect_to', urlencode($redirect), $loginUrl);
        }

        return $loginUrl;
    }

    public static function modifyLostPasswordURL($lostpwUrl, $redirect="") {
        $lostpwUrl = wp_login_url() . '#lostpassword'; // Link to lostpassword URL

        if(!empty($redirect)) {
            $lostpwUrl = add_query_arg('redirect_to', urlencode($redirect), $lostpwUrl);
        }

        return $lostpwUrl;
    }

    public static function modifyRegisterURL($registerUrl) {
        if(!is_user_logged_in()) {
            if (get_option('users_can_register')) {
                $registerUrl="<a href="" . wp_login_url() . '#register" class="btn">' . __('Register') . '</a>'; // Link to register URL
            } else {
                $registerUrl="";
            }
        }

        return $registerUrl;
    }
}

First we need to filter the output of the function wp_login_url() which is used by wp_login_form() in the forms action attribute.

Look at the method modifyLoginUrl(). Here we store the URL of the page login inside the variable $loginUrl. This page must exists inside WordPress, so create it first.

Next we need a filter for the function wp_lostpassword_url() and wp_register(). It’s basically the same. The URL of the site is stored inside the variable $lostpwUrl and $registerUrl.

Last but not least, you have to call the class on the right action hook. For themes this should be after_setup_theme and for plugins you can use plugins_loaded.

Themes / functions.php

add_action('after_setup_theme', 'wpse29338_admin_init');
function wpse29338_admin_init() {
    WPSE29338_Admin::setup();
}

Plugins

add_action('plugins_loaded', 'wpse29338_admin_init');
function wpse29338_admin_init() {
    WPSE29338_Admin::setup();
}