Jon actually has a number of things that are similar to Rhaegar but just not physical similarities, but Rhaegar's daughter didn't seem to have many physical similarities to him either, so assuming Rhaegar is his father, it isn't that Jon has nothing of his father in his make-up. His build is a lot like Rhaegar's when described in one of Bran's early chapters (I believe) - he (or someone) is comparing Jon and Robb and points out that while Robb is stocky, Jon is more nimble with a different frame. There is also the melancholy personality, which Rhaegar was said to have, and the flashes of temper Jon shows at various points in the story.
I think you've probably got a good assessment of the state of the relationships. She liked Renly well enough but also wasn't deluded enough to think she would find romantic love there, that said I think they would have had a decent enough marriage if it had lasted. She definitely hated Joffrey, there were too many times where she'd smile when she was with him but still have that look in her eye that he was totally disgusting and she was grinning and bearing it. Though I think she's open to having a good relationship with Tommen down the road, at this point she's manipulating him but not just to ensure her own place but to get him out from under Cersei control so he doesn't become another Joffrey. I.e while it is definitely manipulation, it is manipulation with an eye towards him being a decent human being and good king so not just for her own ends. That said, I still don't think it will end well for her but that has more to do with the likelihood of Tommen keeping the throne than anything she's doing on her own.
I think it's likely that she either goes to the wall or my preference is that she goes in search of Rickon. The one positive thing that I found in her storyline last season is that she was told by Theon (i.e. not some just some rumor) that Bran and Rickon were alive. Sure Jon knows about Bran from Sam but 1) Jon has his own problems to deal with and 2) doesn't know about Rickon. Since Rickon was sent specifically to the Umbers (99% sure it was them) by Bran and we know Rickon is back next season, it would seem logical that Sansa head in that direction since she's fully aware that her brothers are alive and fled somewhere in the North. The best case is that she and Theon meet up with Brianne and she escorts them in their journey which ends with her finding Rickon and getting the North to rally behind him and her. I do think LF might play into that with his army, but I'd much rather Sansa have the North behind her on her own, than have her just go running back to LF and potentially dropping the brother-bomb in his lap without actually having found either of them.
Aerys had that notion however (specific to the North as opposed to everyone which is why he went to Harranhal) because Brandon came to Kings Landing and made a scene. Given that Aerys called for Ned's head as one of the events that kicked off the war I think the survival of House Stark was dependent on Aerys not being on the throne in the end regardless of whether Lyanna and Rhaegar were together in a willing way. My comment was more for the period before Brandon showed up in Kings Landing, after Brandon's arrest Lyanna's situation was really a non-factor in the future of House Stark. Had the heir to House Stark not threatened the Crown Prince, I really don't get the impression House Starks' place was in the kind of jeopardy that it was in once it started harboring a child of House Targaryen from the new king. If anything I think Aerys would have liked a northern hostage to keep the North in line much like he kept Elia close to control Dorne.
As for why Ned didn't speak too ill of Aerys, I think Ygrain has the right of it, Ned recognized the insanity and while he knew that meant Aerys couldn't be trusted to rule, it didn't require the same sort of lingering grudge because 1) the man was crazy and 2) dead.
Agreed, but if Lyanna did choose to break her betrothal, those actions wouldn't have resulted in any punishment for House Stark or the possibility of them losing Winterfell at the time, given that Robert was just a lord and Rhaegar was the crown prince. That possibility only arose once House Targaryen was dethroned and Robert became King. Given that Ned likely chose to protect Lyanna and the family by proxy, instead of turning Jon over to Robert as Rhaegar's son (which he likely is regardless of whether they got married and he was legitimate) suggests that he didn't fault Lyanna. From the brief moments we have where Ned thinks of his sister, I get the impression there was genuine love there with no qualifiers, i.e. Ned wouldn't have thought differently about Lyanna if she did in fact ditch Robert or if she was taken. Granted that could be because she died in the end, but either way the end result is the same where he doesn't fault her beyond thinking she was a little reckless.
1) I've thought since I first read it that Dany will see Drogo again, but mostly that he'll come to get her as she dies herself after leaving an heir (who I hope isn't Jon's - if Aegon can serve any purpose it is to get some fresh blood into the Targ line - and no I don't think he's Rhaegar's son even if he may have Targ blood).
2) Jon being left with the responsibility of a Kingdom (as I think Dany will leave an heir - I see Jon being more of a regent than a crowned king), especially when by everything we've read he wouldn't really want it, would never have thought of it for himself, but is totally the kind of person to step up because it is his duty and it needs to be done, would be a bitter sweet ending for him cause at best the Game will pause for the Others, it won't stop.
By what the characters know, Robb is the older of the two but not by much since Jon is supposed to have been conceived after Ned married Cat - thus why Cat thinks he broke his vows but was understanding about it - they were just married, virtual strangers, Ned was off at war, etc.
I think though since GRRM said that Jon was born around the time of the sack of KL, Jon would be a little bit older than Robb though not by much that one could tell by looking at them.
I get the whole Tyrion thing, though I think the evidence points more strongly to potentially Cersei and Jaime being Aryes' since the only contact we're sure of happened during the bedding and they are older than Tyrion. True something could have happened later but a large part of the reason I don't like it is it would validate Tywin's dislike of Tyrion and his determination to never have him inherit. While I'm no fan whatsoever of the idea that any of of Tywin's kids are fathered by Aryes, it would at least be more satisfying if it was the children he adored vs the one he hated for his whole life.
Any theory where any of the Lannisters are Targs. I see where people get the evidence they use to back it up, just don't find it that convincing, or see why it would be an addition to the story (I think the dragonriders are Dany, Bran,and Jon). Where as I can see where it would be an addition to the story were R+L=J to be true, which seems likely. Whether he's legit or not, I do think Targ blood may give him an in with dragons, and if he plays a part in saving the kingdom (I'm of the belief it won't be just one person), the true legitimacy may not matter too much to the characters (I do think the readers will get a definite answer). I get why people who like Jon wouldn't want him to make a claim for the throne or would want him to stay with the NW, but 1) I think Jon will be in play for the Iron Throne precisely because it would not make him happy and not be something he seeks out by himself, 2) the whole FTW scene makes a good case for his not being in the NW anymore (either by choice cause who wants to stay in a place where people literally stabbed you in the back for essentially being farsighted enough to see the big picture or cause they don't want him anymore given that he should be dead) and 3) I'm not convinced the Wall or NW will exist in a way that it is place for Jon to be at the end of the story.
The Others are something other than evil.
Dany will go mad at the end. I don't see it, mind you I don't see her living through the series either, but I don't think she'll be crazy when she dies.
Haven't seen it in this thread yet but have seen it on the boards several times...the idea that at the end there will somehow be a representative government ruling Westeros when all is said and done.
ETA: That Jon dies and is resurrected. I'd much rather he just come really really really close to death like Bran but not cross over so as to avoid the whole undead not really alive thing that happened with Beric and Lady Stoneheart.