Is it permissible to adopt an illegitimate child and give him the family name, provided that he is informed of the truth when he grows up? And does he have the same reward as an orphan?

Adoption is absolutely forbidden, whether the child is a foundling, an orphan, or otherwise. As for caring for and raising a foundling, we hope that the person will be rewarded with the reward for sponsoring an orphan, because a foundling's parents are unknown, and he has the ruling of an orphan. Sponsoring him is a communal obligation (fard kifaya) and earns the reward of the one who sponsors an orphan.