If booked on Hilton.com I never bring a print-out. If booked on third party webpage you often have to print out and bring a voucher (it will be stated when you make the reservation) - then I will obviously do that. Anyway, I would be able to retrieve the confirmation email if this issue would arise at a hotel - it has been neccessary a few times when the hotel would like to charge a different rate.

I have never experienced that the hotel ask for the cc that was used for the prepayment when the booking was made. Either the payment went through or was rejected after making the booking - in the latter situation you will get an email asking for updated cc details or the reservation will be cancelled.