Does my Daughter has Custody of my granddaughter?

I'd say so, yes, because there is no custody agreement in place.

if he contests it, or if she wants any child support, then she would need to go to the courts to get a sole custody and child support agreement in place.

If the father has not been established through the courts then she is the sole custodial parent until then. I strongly suggest she does establish him though, for visitation and child support. (though you can get child support without the courts)

No she does not have sole custody until there is something filed in the courts. No one has custody of a child regardless if their married or unmarried. There has to be an agreement.

I've been in this same situation. just because it doesn't say he is the father, he is recognized as the father, so he has the same rights as you do. so get your papers filed and get them filed fast!

It is different in each state go to http://www.nationalfamilysolutions.com/custody.html that is for Missouri.

The mother has custody of her child, but she does not have legal custody though the court, what this means is if the father is on the birth certificate he has rights as well, and although he may not be able to get custody of the child he can file for joint custody and visitations, if the mother files with the court before the father is able to do this then she does not have to worry about the father later down the road to threaten her with custody, the most he could do is file for visitation, and that is if he is on the childs birth certificate, if not then he would have to take a DNA to prove parenty prior to filing for visitation, so it is always good to go though the court. Hope this makes sinse to you, but it is the law and for your daugters protection.