1. Rhaegar and Lyenne never married = Jon is bastard child and can not ever be Targaryen. 2. Although being Snow, Eddard raised Jon as his own child, thus he sees himself as Stark and sees Ned as his father - learning who his real parents were won't change that. 3. Everything he accomplished is under the name of Jon Snow, I don't think he's gonna change it. 4. He will never become King, even if he is one of the three heads of Dragon, in best case he will remain Lord Commander of the NW. 5. Hmmm... maybe Lord Commander Jon The Warg Snow. BTW, I also think that Viserion is Jon's Dragon.