Hi,
The customer will receive the notification emails when the order status changed (and if you change it in the backend, you have to check the box "notify customer" to send him automatically the email).
The invoice is available in the front-end, there is a link in the email to access it.
When the order status allow if (there is an HikaShop option to set that), the user will be able to click on the button "invoice" while seeing his order in the front-end.
But the invoice is not sent by email (the email display the summarize of the order but does not act as an invoice).
The plugin "PDF invoice" allow you to generate a PDF (when the order status authorize it) and attach it to the order status notification email.
Regards,
Jerome - Obsidev.com
HikaMarket & HikaSerial developer / HikaShop core dev team.
Also helping the HikaShop support team when having some time or couldn't sleep.
By the way, do not send me private message, use the "contact us" form instead.