Hi,
If you activate the "order creation notification", then the customer will receive an email notification when their order is created.
Also, when the order status is changed to "confirmed" by the payment plugin you're using following a payment notification from the payment gateway, the "order status notification" email will be sent.
Regarding the sending of the "order status notification" to the customer when the order status is changed to "pending", it depends on how the payment plugin is coded. In the payment plugins we develop, we usually don't request HikaShop to send any email to the customer.
Regarding the admin, he will receive the "order admin nofication" email when the order is created. And when a payment notification is sent to the payment plugin by the payment gateway, a "payment notification" email will normally be sent to the admin, whether the notification is a success or pending or an error so that the admin knows what's going on and can act upon it.
From what you're saying, it seems that the orders status is never set to "confirmed" by the payment plugin, indicating that it doesn't receive a success payment notification from the payment gateway. So I think your problem is not at the email configuration level but first at the payment method level.
For example, for the PayPal payment plugin, there is a "enable validation mode" setting in the payment method. If you activate that setting, then you need to log in your PayPal merchant account to validate the payments so that PayPal can send the payment notification to your website and the order can be confirmed, upon which the customer will be notified with the "order status notification'. If you want that to be automatic, then you want to turn off that setting, like it is by default.
But I don't know which payment methods you're using, nor how you configured them, so I can't say much more.