Page template doesn’t display simple HTML if inserted into other page?

Couldn’t you use the custom form like this:

STEP ONE:

Open a Text editor and paste your form content-

<?php
// Custom Contact Form
?>
<div id="my-ccf">
<form id="contactform" method="post" action="<?php bloginfo('template_directory'); ?>/_assets/submitemail.php">
    <input name="user"  type="text" value="Name" />
    <input name="email" type="text" value="E-Mail"/>
    <input name="url" type="text" value="Website" />
    <input name="subject" type="text" value="Subject" />
    <textarea rows="10" name="message" cols="50" id="message">Your message here...</textarea>
    <input type="submit" class="button" value="SEND" id="send" />
    <div id="error"></div>
</form>
</div>

STEP 2:
Save the file as my-contact-form.php (upload my-contact-form.php to your themes root directory)

STEP 3:
When you want to display your contact form, place this in the template file:

<?php get_template_part('my-contact-form.php'); ?>