Can you get them anytime, or do you HAVE to start a new game after watching the scenes? I rather not start a new game, because I've already got most of the cards on this game. Help, please. Thanks.

I don't know.....I watched every scene before playing CoM and didn't get any card

I beleive you have to get all the trophies from 358/2 days (which means yes, you have to watch it along with other things to get the trophies).

Keep in mind, I have absolutely no idea.  However, I'm going to take a logical guess in thinking that if you watch the Days scenes in the middle of playing Sora's story for Re:CoM, you will be able to access them.  I believe this is the case because in Re:CoM, once you beat Riku's story, you can go back to Sora's story and get various cards such as the Ansem Enemy Card.  So it seems likely that you'll be able to do the same with the Days cards.

Nope. I'm pretty sure you need all the Days trophies before making a new game. 

Chain of memories? you mean organization XIII Cards to get for the game right? beat game as Sora

just for good measure you must beat game with Riku (Any mode should do)

 

After that go to the worlds you visited in your memories and obtain the gold card treasure key called  room to reward or somethin (you can obtain this card in any world as long as you dont mind putting in the hours to get it thru lotsa fighting) then go find the room i know

 

Ansem Card- Castle Oblivion.

Axel-obtain thru play

Demyx-Atlantica
Xigbar-Hollow Bastion
Larxene- obtain thru paly
Lexaeus-Castle Oblivion.
Luxord-Agrabah 

Marluxia-obtain thru play
Roxas-Twilight Town
Saix -Traverse Town
Vexen-obtain thru play
Xaldin-Monstro
Xemnas-Wonderland
Zexion’s-Card-Destiny Islands

You can watch the cutscenes whenever you want and still be able to get the cards in save files you made before hand. Trust me, I got them all after completing Re:CoM before 358/2 Days.