I can't speak specifically for UA, but generally the airlines have your CC information in your FF profile. So a reservation linked to your Mileage Plus number would indicate that you have the card.

More often than not they are a little lax about updating profiles. For example, if you are elite on an airline at the time you book a flight, you can usually get the preferred seats, and they don't take them away if you drop out of elite before the flight date. But you won't get upgraded if the upgrade window is when you're not elite anymore.

You certainly shouldn't need the card in your physical possession. As to what happens if it's been cancelled, I'm not really sure.