Magento – Newsletter signup not showing on one page checkout

You are missing the inline javascript code that defines the billing variable. Compare the billing.phtml files in your current theme against the ones in:

app\design\frontend\base\default\template\checkout\onepage\billing.phtml

app\design\frontend\base\default\template\persistent\checkout\onepage\billing.phtml

You should see similar to below:

<script type="text/javascript">
//<![CDATA[
    var billing = new Billing('co-billing-form', '<?php echo $this->getUrl('checkout/onepage/getAddress') ?>address/', '<?php echo $this->getUrl('checkout/onepage/saveBilling') ?>');
    var billingForm = new VarienForm('co-billing-form');

    //billingForm.setElementsRelation('billing:country_id', 'billing:region', '<?php echo $this->getUrl('directory/json/childRegion') ?>', '<?php echo $this->__('Select State/Province...') ?>');
    $('billing-address-select') && billing.newAddress(!$('billing-address-select').value);

    var billingRegionUpdater = new RegionUpdater('billing:country_id', 'billing:region', 'billing:region_id', <?php echo $this->helper('directory')->getRegionJson() ?>, undefined, 'billing:postcode');
//]]>
</script>

Can't Comment, (not high enough reputation), but the function that is resulting in the error is in the opcheckout.js script, line 372, where the Ajax function is called that saves the billing information.

I would start by window.console.logging all of the parameters that are passed to that function before it is called so that you can diagnose the issue. My guess is that one of them has problems (is null or of the wrong type based upon your error).

When you revert to the base theme, does the problem still occur?

If not, do a comparison between the template files (likely package/template/checkout/onepage.phtml and/or package/template/persistent/checkout/onepage/billing.phtml) with their base package counterparts to locate the differences, and hopefully the issue.