How do you think that this case will most likely work out?

It is a custody and paternity case. The father is requesting the judge order a paternity test and then pending the results, award him shared custody. He has not been allowed any contact with his daughter since birth and isn't even on the birth certificate (thats why he has to establish paternity)
The mother's case consists of
1. He walked away.
2. He never wanted her
3. He has threatened to kidnap her
4. He has physically injured the mother herself
5. He has never paid child support
6. He has never contributed to her welfare
Now the truth is that the father did not walk away. The mother, in fact, threatened to get a restraining order on him because he wouldn't stop contacting her about seeing his daughter. He has never put his hands on the mother and there is no proof that he ever did. He offered to pay child support, but she refused to file because it meant he would be able to get a paternity test. He has also dropped off gifts and such for the little girl, but they have been thrown away. The entire contents of the mother's case are either complete fabrication, or half truths based on situations out of the control of the father but created by the mother. The father is planning to make it plain that he has tried. He has emails and letters documented between himself and his ex. Also between his ex and his fiance, his parents and his ex's parents, and his fiance and his ex's parents. They all show that the mother is only doing this because she wants revenge on him, although her need for revenge is never clearly stated. She just says that she hates him so he can't see their daughter. The father has sought legal aid and is getting a shared parenting plan drafted as well as his arguments solidified so that he has the optimal chance of being awarded at least visitation. Does it seem likely that the judge will see through the mother's lies and give the father parental rights?

Oh and he never threatened to kidnap his daughter either. That was complete fabrication and has no proof whatsoever.

Once paternity is established the father will be ordered to pay child support and visitation and custody will be settled. There doesn't appear to be any reason for the father to be denied visitation from the info you provided.