But like I said if y'alls theory is right than if anything Jon is a Targaryen. Even if his mom is Lyanna Stark he's still Targaryen, you wouldn't classify Robb as a Tully would you?
Well Robb definitely has Tully traits, and tends to act more like his mother than his father at times, to his peril.
I absolutely think being born of a Stark woman and any other house's father would qualify them to enter the crypts.
It's the blood, not the name that makes the man. Why else would Ned have protected JS. He didn't have the Stark name, but he was of his blood and a Stark nonetheless.