did lelouch gain immortality from his father and if he did how did he gained immortality??

Actually, Lelouch is dead. There is a lot of controversy about this topic but the truth is that Lelouch died. The director of Code Geass even berkata that Lelouch's body was kept in Damocles and sent to the sun.

Also, in the end, CC looks to the sky while 'talking' to Lelouch.

Proof that Lelouch is dead.

This is the most hotly debated topic in all the fandom. No one knows the answer, though; There are all sorts of arguments on either side. But there is no answer, just opinions

No lelouch is dead there were no proven he is alive how can he get charles brittania geass?