Do I need to pay to ride the train from T2 to T3 in the Beijing airport?

No, it is free.

Do I still need to purchase a card ticket? I have heavy luggage to transfer to domestic flight. Would train be accessible from within the building of T3 be any easier than the inter terminal shuttle? I assume I'll need to exit the building to catch the shuttle?

Actually, it is free shuttle buses between terminals. Just get on the bus and no ticket/card is needed. At T2, you can take that out of Door No. 7 in the 1st floor. As I know, it will take you directly to the departure floor of T3.