Will I Be Billed for Both the Bandwidth Used by the Load Balancer and the Bandwidth Used by Backend Servers?

This depends on your services. If backend servers are only accessed from within a VPC, you do not need to bind an EIP to each backend server and assign bandwidth because requests from the clients are received and routed to backend servers by the load balancer. You only need to bind an EIP to each backend server if they need to provide services accessible from the Internet. In that case, you need to pay for the bandwidth used by your load balancer and also the bandwidth used by the backend servers.