Datepicker and custom contact form (custom wp plugin)

thank you for the help in advance. I have been trying to find tutorial about building my own contact form for wordpress (without plugin). I thought i found one (part of the question).

    <?php
/*
Plugin Name: Self Catering Contact form plugin
Plugin URI: http://example.com
Description: Simple non-bloated WordPress Contact Form use [contact_form]shortcode
Version: 1.0
Author: Agbonghama Collins
Author URI: http://w3guy.com
*/


function html_form_code() {
    echo '<form action="' . esc_url( $_SERVER['REQUEST_URI'] ) . '" method="post">';
    echo '<p>';
    echo 'Your Name (required) <br />';
    echo '<input type="text" name="cf-name" pattern="[a-zA-Z0-9 ]+" value="' . ( isset( $_POST["cf-name"] ) ? esc_attr( $_POST["cf-name"] ) : '' ) . '" size="40" />';
    echo '</p>';
    echo '<p>';
    echo 'Your Email (required) <br />';
    echo '<input type="email" name="cf-email" value="' . ( isset( $_POST["cf-email"] ) ? esc_attr( $_POST["cf-email"] ) : '' ) . '" size="40" />';
    echo '</p>';
    echo '<p>';
    echo 'Subject (required) <br />';
    echo '<input type="text" name="cf-subject" pattern="[a-zA-Z ]+" value="' . ( isset( $_POST["cf-subject"] ) ? esc_attr( $_POST["cf-subject"] ) : '' ) . '" size="40" />';
    echo '</p>';
    echo '<p>';
    echo 'Your Message (required) <br />';
    echo '<textarea rows="10" cols="35" name="cf-message">' . ( isset( $_POST["cf-message"] ) ? esc_attr( $_POST["cf-message"] ) : '' ) . '</textarea>';
    echo '</p>';
    echo 'Select date <br />';
    echo '<p><input type="text" id="datepicker" name="datepicker" value="' . ( isset( $_POST["datepicker"] ) ? esc_attr( $_POST["datepicker"] ) : '' ) . '" size="40" />'.'</p>';
    echo '<p><input type="submit" name="cf-submitted" value="Send"/></p>';
    echo '</form>';
}
function deliver_mail() {

    // if the submit button is clicked, send the email
    if ( isset( $_POST['cf-submitted'] ) ) {

        // sanitize form values
        $name    = sanitize_text_field( $_POST["cf-name"] );
        $email   = sanitize_email( $_POST["cf-email"] );
        $subject = sanitize_text_field( $_POST["cf-subject"] );
        $message = esc_textarea( $_POST["cf-message"] );
        $datepicker = sanitize_text_field($_POST["datepicker"]);

        // get the blog administrator's email address
        $to = 'atesz@atesz.co.uk';

        $headers = "From: example@yourdomainname.com" . "\r\n";

        // If email has been process for sending, display a success message
        if ( wp_mail( $to, $subject, $message, $headers ) ) {
            echo '<div>';
            echo '<p class="red">sent</p>';
            echo '</div>';
        } else {
            echo 'An unexpected error occurred';
        }
    }
}
function cf_shortcode() {
    ob_start();
    deliver_mail();
    html_form_code();

    return ob_get_clean();
}

add_shortcode( 'contact_form', 'cf_shortcode' );

This is what my contact-form.php looks like. My first problem that it doesnt send the email - or it doesnt arrive even tho it says that the mail was sent. I have read a few things for example some hosting providers dont allow you to use different email address to your domain ones... Could somebody explain this?

If you still have enough patience left, could somebody guide me through how to implement datepicker (which i thought i have done), and how to handle or get the data from it to be included in the email?

Thank you very much for your time...