For all my Easymode forms I want to automatically fill in the name and Email fields with those of the currently logged in user

With EasyMode forms, the PHP code is different, and has to be entered in the Form Settings > Scripts > Init Scripts section. Here is an example of what you would need to enter: 

<?php
echo '
var _name = "'.JFactory::getUser()->get('name').'"
var _email = "'.JFactory::getUser()->get('email').'"
var _username = "'.JFactory::getUser()->get('username').'"
';
?>

function ff_YourFormName_init()
{
     ff_getElementByName('name').value = _name;
     ff_getElementByName('email').value = _email;
     ff_getElementByName('username').value = _username;
} // ff_YourFormName_init

Make sure you change the YourFormName part of the code above to the actual name of your form.