Hi Patrick,
Yes we can select the serial numbers manually or automatic in SO.
Yes we can view the Serial number or Batch number on the Packing slip and Invoices. For this we need to do small setups in Accounts Receivable. Goto AR > Setup > Forms > Forms setup …> go to Packing slip / Invoice (tabs) > Show in sheet field group … You can select the Batch or Serial numbers. It will display the Serial or Batch numbers when we do Packing slip or Invoice.