Re: Jon being a legal heir - The World of Ice and Fire goes into more detail about the legality of Targarian "plural" marriage - and basically, it's never been really accepted in Westeros even when the Targs ruled completely. So even if Rhaegar and Lyanna married each other legally, their son wouldn't be generally acknowledged as having any claim to the throne or anything else. And bringing up Tommen is a good point, except that there is a world of difference between being an acknowledged heir in the line of succession, rumored to be a *******, and being an acknowledged and named ******* claiming to be a trueborn heir to the throne.
But really, none of this matters. This isn't a story where Jon shows up claiming to be a bona fide prince and everyone goes "Yay! The Game of Thrones is over!! Here is our king!!" Rhaegar Targariian HIMSELF could show up and he would still have to fight for the throne at this point. I can't believe anyone whose read/watched this far thinks this thing gets concluded that cleanly. It may be a marriage in the end - Elizabeth of York did marry Henry Tudor after all - but only after a massive amount of fighting between everyone and after all the other major players are dead. Not to mention the Walkers are going to be involved which changes everything.
I honestly think that finding out about R+L=J won't affect the Iron Throne one bit. It will affect Jon, and hopefully bring him a measure of peace about who he is and how dearly Ned protected him all those years, and it may help him recognize some unrealized magic that the mingling of two ancient, magical bloodlines gives him, from dragonriding to being a savior or a sacrifice. But if Jon ends up on the throne - married to Dany, or anyone else - it will be because of what he does against the Walkers, not any type of genealogical claim.