Programatically retreive page content from its template page

You can get part of the way there by querying the page template using get_page_template_slug() inside your loop. For example, you can do something for non-default page templates:

if ( '' != get_page_template_slug( $page->ID ) ) {
    // This page uses a custom page template;
    // do something
}

Or, you could look for your contact-form page template specifically:

if ( 'template-contact-form.php' == get_page_template_slug( $page->ID ) ) {
    // This page uses the contact form custom page template;
    // do something
}

The problem you’ll encounter is the do something part, because you’ve hard-coded your contact form into the template.

To do what you’re wanting to do, you’ll probably need to convert your contact-form code into a shortcode or Widget, so that you can parse/output it in your custom loop, such as:

if ( 'template-contact-form.php' == get_page_template_slug( $page->ID ) ) {
    // This page uses the contact form custom page template;
    // parse the shortcode
    echo do_shortcode( '[contact-form]' );
}

…or:

if ( 'template-contact-form.php' == get_page_template_slug( $page->ID ) ) {
    // This page uses the contact form custom page template;
    // output the widget
    the_widget( 'my-contact-form' );
}

(Note: shorthand code for do_shortcode() and the_widget(); refer to Codex for proper usage.)