North Star

Joffrey kissed her in a very Princess Bride-y scene back in season 1, right before everything went to hell for her. She also married Tyrion, but I don't know if Westerosi wedding ceremonies include a kiss or not. We didn't see one, anyway.

Lysa's still with us, so I think the champion's death is all that matters.

Tywin explicitly said Sansa was heir, though, and identified Margaery in the same terms. A woman's husband in a time like this could, of course, exert considerable influence through his marriage, especially in Sansa's hypothetical case, where she was a prisoner and the point of the marriage was to effectively transfer her powers over to Tyrion. Or, as we subsequently found out, to her and Tyrion's adult son, since Tywin apparently wasn't any fonder of letting Tyrion run the North than letting Sansa do it. Actually, that raises a new question about what Tywin's plans for the North going forward are now. Bolton was named Warden temporarily, with the understand that the Lannisters would get it once Tywin had raised somebody he trusted to take up the job (nobody ever raised the issue that there was like a 50/50 shot that any child Tyrion and Sansa had would also be a dwarf, though I suppose they could have just kept producing them until one who wasn't was delivered), but that would seem to be out the window at this point.

Tywin in episode 3.05: "The Young Wolf has lost half his army; his days are numbered. Theon Greyjoy murdered both his brothers. That makes Sansa Stark the heir to Winterfell." Even if he hadn't said that, what else would "the key to the North" mean apart from being heir? Likewise, Margaery has been identified as Loras' heir; hence Tywin's coercing Loras into marrying Cersei by threatening to name him to the Kingsguard otherwise, which would make Margaery heir, and hence her children by Joffrey, which would merge House Tyrell into "House Baratheon" in the long-term.

They haven't said whether there are other Arryns, but especially given that the female line can inherit (see: Sansa, who most people think is Lady of Winterfell right now, with Tyrion the Lord consort or whatever you want to call it; and Margaery, who is apparently next in line to Highgarden after Loras, per last season), you would think there'd be somebody else out there in the succession. But these noble families are all as big or as small as the plot requires, so I guess we'll see. It's possible Robin is the last person in the blood succession, which would probably lead to the Eyrie being vacant and the crown able to name whomever they want to the position.

Lysa is the regent for her minor son, so if Robin died, the Eyrie and the Vale would pass to whoever the next-closest Arryn is.

It doesn't strike me as unreasonable that she'd believe her aunt wanted to help her. Catelyn wasn't aware that Lysa was completely bonkers until she visited her, so presumably Sansa would only have ever heard positive things about her.

Littlefinger, apparently.

He referred to having sex while on duty, I believe, not deserting. The former doesn't meaningfully impact the strength of the Watch, whereas one gets the impression that much of the Watch's membership would pick up and leave if there was no penalty involved.

Sansa's pretty thoroughly implicated already by fleeing just as the murder happened.

Littlefinger pretty much confirmed his involvement, since he knew Joffrey was dead when Sansa got there, despite nobody telling him that. Unless he's got a cell phone on that boat.

We saw Ned as Hand passing sentence on the Mountain by himself, which would fit with the standard medieval justice model. I would imagine both the Hand and the Regent would have the authority to sit in judgement; given that Tywin is Hand, it probably doesn't matter who the Regent is (if it Cersei I imagine she'd want that job herself, but again, I doubt Tywin would cede that to her).

Without challenging the general theory (the photo evidence is good), Renly supposedly was the clear favourite to win at the time of his death (if Littlefinger looking to jump ship is any indication), so I don't think his death put the Tyrells in a better position. They just had to scramble to return to the same position they were in before. I don't believe Joffrey's death leaves the throne to the Tyrells. Robert's death didn't leave the throne to Cersei. The new king would be Joffrey's brother (who turned up for the first time in forever, albeit I don't think he had any lines), and his regent would presumably come from his own family, not his sister-in-law's. I wouldn't be surprised if Tywin takes that job himself this time, unless there's some rule against the Hand also being Regent.

Margaery would only be in a position to be regent for one of her own children, I believe. With Joffrey dead she's pretty much back to square one. Cersei, as Tommen's mother, would seem the more likely regent, though Tywin might prefer to just take that office himself this time and dispense with pretenses. On the whodunnit front, there are some superficial reasons to make me wonder about Oberyn, but on the whole I think probably not. The main thing would be that the reappearance of Sansa's drunken knight/fool coincides pretty much exactly with Oberyn's arrival, and they made a point of showing that Oberyn actually got into the city without anybody noticing in the premiere. The lead Dornish guy Tyrion spoke to said he had arrived earlier that morning, but there's no actual corroboration of that, so he could have been there for days or weeks, for all we know. Weighing against it are Oberyn's seeming remoteness from the murder scene (though proxies are always a possibility there, among the servants), and his being so openly antagonistic, which would only draw suspicion to himself (unless it's a double-bluff). If it was him, though, while his motive for wanting Joffrey dead would be blatantly obvious, I'm not sure what he'd want Sansa for. Apparently his sister's husband ran off with Lyanna Stark and he's not pleased about that, but it seems like a stretch that he'd be looking to settle scores with her niece who wasn't even born then. Could be the same political opportunism why everybody wants Sansa, I suppose. Or he could want to help her, but when has Sansa ever been that lucky?

Loras may be gay, but he'd be a better choice than Tyrion, in that if she were married to Loras she would get to leave and be under the protection of House Tyrell.