Problem using wp_mail function

There are a couple of problems that may lead to failure of your script.

First, you have a check for $mailSent at the top so you can redirect. However, at that point, it is undefined. It would be less likely to be a problem if you move it to where you check it at the end and set an error. In fact, I would move all of that outside your if (!$suspect && !$missing && !$errors) { check.

And that brings up another similar problem. Where does $suspect come from? This appears to be undefined as well. I'd take that out if it isn't supposed to be there.

Here are those changes:

$errors = array();
$missing = array();

if (isset($_POST['send'])) {
    $to = '[email protected]';
    $subject = 'Form';
    $expected = array('name', 'email', 'pick_up_adress', 'drop_off_adress', 'phone', 'pick_date', 'pick_time', 'type');
    $required = array('name', 'email', 'pick_up_adress', 'drop_off_adress', 'phone', 'pick_date', 'pick_time', 'type');
    $headers = "From: Metro<[email protected]>\r\n";
    $headers .= 'Content-Type: text/plain; charset=utf-8';
}

// Array proccesing
$mailSent = false; 
if (!$missing && !$errors) {
    $message = '';

    foreach($expected as $item) {
        if (isset(${$item}) && !empty(${$item})) {
            $val = ${$item};
        } else {
            $val = 'Not selected';
        }

        if (is_array($val)) {
            $val = implode(', ', $val);
        }

        $item = str_replace(array('_', '-'), ' ', $item);
        $message .= ucfirst($item).": $val\r\n\r\n";
    }

    $message = wordwrap($message, 70);

    // simple php doc $mailSent = mail($to, $subject, $message, $headers); 
    $mailSent = wp_mail($to, $subject, $message, $headers);
}

if ( $mailSent ) {
    header('Location: after_form.php');
    exit;
} else {
    $errors['mailfail'] = true;
}

Lastly, I would be inclined to put this in an action hook rather than in a template. It can certainly work in a template, so that's up to you, but I think a hooked action such as init would be more portable.