How do I stop the password from getting emailed?

The emails can be edited via the menu System->Emails which is available in the business edition of HikaShop.

You cannot stop this email from being sent by just editing the email with the current version. It will be possible in next version by adding the code

$data->email='';

$mailClass->sendMail($mail);

If you just want to remove the password from the email, you don't need to edit the email. You can just edit your hikashop translation file in the config under the tab languages and empty the translation for HIKA_PASSWORD so that the line becomes:
HIKA_PASSWORD=""

Indeed, when a user registers via HikaShop, like when a user registers via joomla, his account will be confirmed in joomla but won't be confirmed in CB because CB has a special table with the user information instead of using joomla's confirmed value.

What you can do is edit the file components/com_hikashop/controllers/checkout.php and add the code:

			if($user_id && file_exists(JPATH_ROOT.DS.'components'.DS.'com_comprofiler'.DS.'comprofiler.php')){
				 $db->setQuery('REPLACE #__comprofiler SET (cbactivation,id) VALUES (\'\','.$user_id.')');
				 $db->query();
			}

$user_id = $class->getID($user->get('id'));

Ah yes.

To remove the full password line of the email, you need to edit the email and then remove the code :
<?php echo JText::sprintf('HIKA_PASSWORD').' : '.$data->password;?>