Some of the blockchain-based cards are divisible. How does that work in terms of gameplay? If I own half a SatoshiCard I can't use it in the game right?
That's not true. You cant buy half a card
You can buy even 1 satoshi of some (old) card

So you should be very careful while selling it on the CounterParty exchange as a bad buyer can spoil it

Recently I had my buy order partially fulfilled (it was Ripple card), so I did not get the whole card - well, it does not appear on the team tab or somewhere else in the game.