It's her birth certificate with his name as daddy.
No. Too obvious. Besides I think they legitimately don't know why Red is so obsessed with her, which is why Keene was planted there in the first place.
It may be proof (a photo) of Red hastening her adoptive Dad's demise though. That would certainly drive a wedge between her and Red which is probably something "they" want to do since Tom is no longer around to keep an eye on her.