To use the Decks containing Forbidden cards, you must be using a banned card in your deck (to do this, you must have cleared the game with all Page 1 characters).
The note "(D-Card)" indicates the card is selected for "Destiny Card" and can is randomly drawn from among all the D - Cards when Destiny Draw is selected. If you have this turned off in options, then the Destiny cards do not apply. You also cannot Destiny Draw a card that you have already drawn/played.
You can also edit their default partner deck when you've already reached the 4th heart event. The note "(F-card)" indicates the card is his/her's favorite card, means that you can't remove it from the deck.