In Sales order when you select another customer in the bill-to field and there IS a registered email on the ship-to customer, the email does not go to the bill-to customer. Instead the email with the invoice is sent to the ship-to customer. 

Since it's the bill-to customer that should pay the invoice, the email on the bill-to customer should be selected. You should alway look at settings on bill-to customer first. Then look at settings on the sell-to customer.