We are pursuing adopting the unwanted child of a friend's sister. A question (among many!): No father on the birth certificate. Father that she says is the father is married with 2 kids and wants NOTHING to do with the child. Another guy, who she originally told was the father and now she says is not, is calling the grandmother (who the child now lives with). If we pursue adoption, and it is agreed upon by the birth mother (and her family, since this is a family we are friends with, and want to remain friends with), do we have to establish paternity and get the approval of the father for adoption?

Yes.

If you do not, the father can come back and get sole custody of the child, as it is his child, and the mother has already signed away her rights.

NOT establishing paternity and having the father sign away his parental rights can in some cases overturn an adoption.

My advice is to get a family lawyer.

We are pursuing adopting the unwanted child of a friend's sister. A question (among many!): No father on the birth certificate. Father that she says is the father is married with 2 kids and wants NOTHING to do with the child. Another guy, who she originally told was the father and now she says is not, is calling the grandmother (who the child now lives with). If we pursue adoption, and it is agreed upon by the birth mother (and her family, since this is a family we are friends with, and want to remain friends with), do we have to establish paternity and get the approval of the father for adoption?
YES. A quick DNA screen will be done but if no DNA established birth father is confirmed then I'm not sure what comes next. You do have an attorney and that person will give you all the right answers. Best wishes.