Hi,
1. An order is technically mandatory.
At the end of the checkout, when you're redirected to the payment gateway, most payment gateways require the shop to send the order id from the shop so that when the payment gateway notify the shop of the payment, the payment gateway give the order id and that way the shop knows which order has been paid.
There is no "sending" of orders. If you're talking about the "order creation notification" email that is sent when the order is created, you can disable it in the menu System>Emails.
2. That's already done by default. When HikaShop receives the notification of the payment from the payment gateway, it changes the order status from "created" to "confirmed" which means that the payment has been received. At that point, the invoice is created and a notification email is sent again to the customer.
3. There is. It is called "confirmed". If you want to rename it, you can change its translation with a translation override:
www.hikashop.com/download/languages.html#modify
Paid is not generic enough. If the order amount is free for example, there is no payment and thus the order is not paid. Regardless you still want to validate it. Hence the "confirmed" status. That's the term ecommerce software usually use by default.
By default, an invoice is not sent with order notifications. This requires a PDF to be generated and added as attachment to the email while the invoice you can print from the backend or the customer can print from the frontend order listing doesn't need a PDF.
If you need that, we provide a plugin specifically for that:
www.hikashop.com/component/hikashop/product/cid-18.html