Is it permissible for a young man to communicate with a non-mahram woman to ascertain a child's lineage?

Adultery is a grave sin that necessitates repentance and abstention from its causes. Opening a woman's Facebook page may be a satanic enticement. It is obligatory to get rid of her page and change the page's URL to block the means (to sin), in fulfillment of the Almighty's saying: "O you who have believed, do not follow the footsteps of Satan. And whoever follows the footsteps of Satan - indeed, he enjoins immorality and wrongdoing. And if not for the favor of Allah upon you and His mercy, not one of you would have been pure ever, but Allah purifies whom He wills, and Allah is Hearing and Knowing." As for a child born out of wedlock, the adulterer has no relation to him. If his mother is married, he is attributed to her husband. If she is not married, he is attributed to his mother. He is a child born of adultery, and the adulterer has no original relation to him.