Magento – How to change reply-to email adress of contact form when using transactional email template

Use handle="sales_email_order_items" instead of handle="sales_email_order_invoice_items" because, some times invoice not be created automatically and you cannot use invoice items in your phtml. Invoice creation related to payment or your store payment configuration. New order email will sent when order state (not status) is changed to processing. Order state will be processing when payment approved or invoice created manually in the backend. See original email templates to figure out the situation.

Your route contacts/index/post - I assume is part of the core Mage_Contacts module in Magento. If so, that template can be configured via Magento admin at System > Configuration > Contacts; see also:

http://www.magentocommerce.com/wiki/modules_reference/english/mage_adminhtml/system_config/edit/contacts

If you were to inspect the controller, you would see it referenced:

# File: app/code/core/Mage/Contacts/controllers/IndexController.php
# postAction, lines ~96-99
....
$mailTemplate->setDesignConfig(array('area' => 'frontend'))
    ->setReplyTo($post['email'])
    ->sendTransactional(
        Mage::getStoreConfig(self::XML_PATH_EMAIL_TEMPLATE),
...

And the constant XML_PATH_EMAIL_TEMPLATE resolves to the configuration path contacts/email/email_template.

If you continue to read this postAction method, you can see that it passes the entire POST data to the template. Therefore, all of its data should be available to you during the transactional e-mail processing:

array('data' => $postObject) // $postObject Varien_Object

So your e-mail template can call in the POST data like so:

{{var data.getFormFieldName()}}

Now, regarding your AJAX response, you may have to override the controller in order to generate a useful JSON response. Otherwise the response is going to be the HTML output of the controller's _redirect call. If you did extend the controller to return your desired JSON, it can be accessed as the first parameter in your callback:

...
onComplete: function(transport) {
    var response = transport.responseText.evalJSON();
    alert(response.custom_response_property);
}
...