Bastards don't have claims to thrones. He has zero claim. If we pretend his parents were married, no one on earth knows or thinks they were married except maybe Bran, and like I said it probably wasn't a legitimate marriage anyway. Neither side wanted it. Robert would have killed both of them if he thought his woman was in the relationship willingly (married or not), and the mad king probably would have too.
I don't think there will be any conflict, unless it's from Dany being paranoid, because Jon is a very reluctant ruler in the first place. (I think she will be thrilled to have a living relative). King of the North is a bit different since it's a brand new title, and also because the Starks were decimated. At one point everyone thought there was no male heir to the throne. They also thought Sansa was lost to the Lannisters forever and Arya was dead. That left literally no one from the strongest house in the North except Jon. So he, a bastards, becoming their ruler was only because of extremely odd circumstances. Also, when Stannis offered to make Jon a legitimate son of Ned, he turned him down.
Agreeing to rule the north was a big leap, and I think Sansa will end up doing that even if Jon survives. Jon wouldn't accept being king of everything. What he would accept, IMO, is ruling the north for Dany, or giving Sansa a strong recommendation and helping bring them to some kind of agreement.
That ending wouldn't be disappointing at all, given all the sh#t everyone's gone through.
From what I have read is that info like marriages, even runaway ones are information that maesters in the citadel keep records of deep in their libraries. And I bet it's one of those out of site out of mind deals. Or at least that's how Reddit seems to think the whole thing probably went down.
Yeah, the r/asoiaf reddit is a deep cave of conspiracies. The maester comment is correct. Logs like this are kept as a record and only really brought back out if there is a legitimate claim. I don't think Jon will want the Iron Throne either. I'm just saying that there is a better than likely chance that R & L got married thus making Jon a more legitimate heir. From what we know, the kidnapping and raping was just a narrative that Robert to legitimize his hold on the Throne. Creating the story that ALL Targs are bad is an easier sell than I lost my girl to a pretty boy.