You will receive Rapid Rewards cards in the mail. If I remember correctly, when you enroll, you can print a card.
You won't have to show birth certificates for the kids. The won't have to show ID at all. I did carry all of our passports with us when we flew in March. DS is 17 (and 6' tall). TSA asked how old he was. When he said 17, they didn't need to see any ID for him.