Does the Muslim father bear responsibility for the actions of the child he had out of wedlock?

Assalamualaikum 1- A Muslim person was having a affair with a Non Muslim English woman, she became pregnant and had a child from him, the English woman has left him 2 years ago, she also took the child with her, the question is that does the child belong to him because he had the child without Nikah(getting Married)? 2-The Muslim person has repented from the sin of adultery,and cut ties with the Non Muslim English woman and the child after she left him but the question keeps coming in his mind that if the child who is staying with his Non Muslim mother also becomes a Kafir will he get punished in the here after for the wrong the child does?