How do you think Jon will react when he finds out that he: 1) a Targaryen, 2) had sex with his aunt, 3) has a stronger claim than her to the Iron Throne?

I think that Jon, who grew up and spent all his life in the north, will insist that no matter who were his biological parents and house - he wholly belongs to House Stark and to the North, and his real father was Ned.

As for sleeping with Daenerys: I think he'll feel uncomfortable, maybe disgust. After all, Jon has always been conservative about interaction with women: didn't have sex with Ros, never visited the brothel at Mole's Town, and rejected Ygritte's advances till he had no choice but to sleep with her (in the book). Having sex with his kin - is not something Jon would have done knowingly.

About the claim to the Iron Throne: I think he'll prefer to be King in the North, and since he cares for Daenerys - will be ready to waive his right in her favor.