Do I have to create an invoice for the customer? Even as a private seller?
The customer is entitled to ask you for an invoice following his purchase. Please send them the invoice by email or by post as soon as possible. The invoice must mention the price of the sale (Selency commission included because you receive the commission invoice on your side) and the price of the delivery if the delivery was at your expense.
If you are a private seller, there is no need to do anything. We can indeed edit for you a sales certificate which will serve as an invoice for the customer. The latter will have the possibility to download into his account!