No, you do not need to print out a copy of your ETIAS approval email nor a screen shot of the online status check showing your approved authorisation. Since the ETIAS is an electronic travel authorisation, relevant authorities and carriers will be able to confirm its validity through their own systems. For your own peace of mind, you can make a note of your approved authorisation number, as well as regularly check the status of your authorisation to ensure it remains valid and up-to-date.