In quest mode, when I complete a scenario, I can rearrange cards between the characters. However, any cards that I move to the lower half of the screen seemingly vanish after I exit the screen.

 

When I click the "View Decks" button in the menu, the decks show up but the "Available" panel on the bottom is always empty.

 

Only when a deck does not have enough cards at the end of a scenario I get to choose from many cards what I want to add in the deck.

 

Is there a way to... banish or otherwise remove cards from my decks at will so I can replace them with other cards that I acquired in previous scenarios?

No, cards not used disappear and are in the Vault. Picture the Vault as the inventory of the world and the decks as the characters' pockets. If you don't actively carry it, then it goes back out into the world to find again.