Does php mail() function work in WordPress

I have an email form in WordPress like this.

html

<form action="" id="contactForm">
        <input type="text" id="contactName" placeholder="Name:">
        <input type="text" id="contactEmail" placeholder="Email:">
        <select id="form-subject">
          <option>Subject</option>
          <option>Accounts</option>
          <option>Bookkeeping</option>
          <option>Start Up</option>
          <option>CIS Returns</option>
          <option>Payroll</option>
          <option>Self Assessment</option>
          <option>Tax Returns</option>
          <option>VAT Returns</option>
                        <option>Other</option>
        </select>
        <textarea id="contactMessage" rows="8" cols="40" placeholder="Message:"></textarea>
                    <p id="submit">Submit</p>
                    <div id="status">
          <p></p>
        </div>
    </form>

jquery

$j('#submit').click(function(){
        //
        var nameVal = $j('#contactForm #contactName').val();
        var emailVal = $j('#contactForm #contactEmail').val();
        var messageVal = $j('#contactForm #contactMessage').val();

        //
        $j.post('wp-content/code/contactEngine.php', { theName:nameVal, theEmail:emailVal, theMessage:messageVal }, function(data){
            $j("#status p").html(data);
            $j("#status p").show().fadeOut(3500);
            if(data.indexOf('Thank You')==0) {document.forms[0].reset();}
        });
    })

php

<?php

      $errors = array();
      $required_fields = array('theName','theEmail','theMessage');
      foreach($required_fields as $fieldname){
         if(!isset($_POST[$fieldname]) || empty($_POST[$fieldname])){
            $errors[] = $fieldname;
         }
      }
      if(empty($errors)){
         $name_field = Trim(stripslashes($_POST['theName']));
         $name = explode(' ', $name_field);
         $firstname = ucfirst($name[0]);
         $email_field = Trim(stripslashes($_POST['theEmail']));
         $message = Trim(stripslashes($_POST['theMessage']));
         //
         $to = "[email protected]";
         $subject = "Email from Website";
         $body = "From: $name_field\n E-Mail: $email_field\n Message:\n $message";
         //
         mail($to, $subject, $body);

             //wp_mail($to, $subject, $body);   

         echo "Thank You $firstname";
      }else{
         echo "Please complete all fields.";
      }

    ?>

The php is in a folder in the wp-content folder so it’s easy for the jquery to access it.

$j.post('wp-content/code/contactEngine.php', { theName:nameVal, theEmail:emailVal, theMessage:messageVal }, function(data){

The variables reach the php and echo back a reply if the form is complete or not.

The actual email isn’t sent – Does the php mail() function work in Worpdress.

I tried the wp_mail() function but this stops the script completely and I don’t get the echo.

I know there are email plugins like Contact Form 7 but I don’t think they will work with this site.

It’s a single page site with a Custom Post Type, I’m not calling any content, which I think I ned to do with Contact Form 7.