I'm just curious, do the trading cards stack between playthroughs?  I'm trying to go for the S Rank, so I know I have to play the game at least 3 times. Do I have to get all the cards in one run, or can I get a few in one run and the rest in another?

They have to be acquired in one run.

well there is also a secret card that you need to get that you don't get through the actual story mode.  you just need to open the right door to find it ;-)

No you can't, but you can reload old chapters to get cards then get the secret card at the end in secrets.  It's not really that hard its just a annoying redoing those missions but I got mine fairly quick.  I would go for the S rank but I'm not up to playing that game two more times after beating it once myself, getting all the cards, watching BR and VJ, I'm done with DP.