Change
<div class="input-box">
<input type="text" name="newsletter_email" id="newsletter_email" placeholder="Email" title="Email" class="input-text required-entry validate-email">
</div>
to
<div class="input-box">
<input type="text" name="email" id="newsletter_email" placeholder="Email" title="Email" class="input-text required-entry validate-email">
</div>
Name should be 'email' if you use standard Magento controller. No need in newsletter_name input then - native controller will not get data from it. If you want to use input name "newsletter_email" and use "newsletter_name" somehow then you should create your own controller.