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.
Are you referring to the sogassets.com exchange. Have you successfully purchased from there or where ever you did before? I bought a card using the Indie counter wallet and it took almost 3 days to come thru. Then I switched to book of orbs and haven't had a problem since