Are invoices automatically generated and emailed to the client?

Yes. Invoices are attached to standard WooCommerce emails, that is:

• Regular invoices are attached to Completed Order email
• Proforma invoices are attached to Processing Order email

Just make sure you have invoicing enabled and "Send by email" checkbox checked. For proforma invoices, you must also check "Enable proforma invoicing" and check "Send by email" under "Proforma Invoice Settings".