Is the young man obligated to acknowledge the girl, and does she inherit from him? And is he obligated to marry her mother, given that he committed zina with her, and she claims customary marriage to him, while he denies the marriage, and then the court ruled for DNA testing?

It is not permissible for the fornicator to claim the paternity of a child born of fornication, because the fluid of fornication is considered wasted and does not establish lineage or inheritance. DNA analysis is not considered a valid proof of paternity for a child born of fornication. The fornicator is not obligated to marry the woman with whom he committed fornication; rather, his duty is to hasten to sincere repentance to Allah.