What if Oberyn won the duel?

[Winterfell is Burning]

Tyrion goes to Dorne.

Tywin is killed by Varys anyway. Maybe he lives longer, but not that much longer. That is, if Oberyn wasn't poisoning him anyway.

Of course, after Tywin dies, Tyrion is the rightful heir to Casterly Rock, but Cersei will try to get it herself, and hilarity ensues.

[The Broke Howard Hughes]

If the mountain publicly implicates Tywin in the rape and murder of Elia, Oberyn kills him and everybody shrugs. His entire plan depended on getting the Mountain to confess and admit Tywin gave the order. With that admission Tywin dies. You see how quickly he was willing to hand over The Mountain to the Martells. Once the Mountain admitted it was him who killed Elia, Tywin knew the guy was a goner. The Martells have an untouched army, and deserve justice for Elia. If Oberyn wins, Tywin goes and nobody steps up to aid the Lannisters in protecting him. The Lannisters could refuse to hand him over but with her daughter as a hostage Cersei doesn't act. Jaime likely won't care, not really. Tyrion says FU, Kevan likely gets murdered anyway. If the Lannisters choose to fight, they get wiped out because Robb Stark showed the world you don't have to fear the great Tywin Lannister. If the fight goes the other way all of Westeros watches with anticipation as House Lannister pays their debts.

[Mithras]

If the mountain publicly implicates Tywin in the rape and murder of Elia, Oberyn kills him and everybody shrugs. His entire plan depended on getting the Mountain to confess and admit Tywin gave the order. With that admission Tywin dies. You see how quickly he was willing to hand over The Mountain to the Martells. Once the Mountain admitted it was him who killed Elia, Tywin knew the guy was a goner. The Martells have an untouched army, and deserve justice for Elia. If Oberyn wins, Tywin goes and nobody steps up to aid the Lannisters in protecting him. The Lannisters could refuse to hand him over but with her daughter as a hostage Cersei doesn't act. Jaime likely won't care, not really. Tyrion says FU, Kevan likely gets murdered anyway. If the Lannisters choose to fight, they get wiped out because Robb Stark showed the world you don't have to fear the great Tywin Lannister. If the fight goes the other way all of Westeros watches with anticipation as House Lannister pays their debts.

What if they got a confession from Gregor? Would it be any better than Robert's will, which was torn apart by Cersei? Tywin would refuse it and there would be another trial by combat at best with an uncertain end. Even if Tywin was found guilty in a trial, he would be allowed to take the black and House Lannister would continue working. Tywin's guilt does not bind his descendants.

One should also note that even if it was revealed that Tywin gave the orders, he could easily say "So what?".

There is no written law that during a sack, women and babies should be spared, even if they are of noble blood.

[The Broke Howard Hughes]

But it gives the Martells just cause to kill him. And simply denying the truth of what The Mountain said won't erase the fact that everyone heard it. Very few people read or saw Robert's declaration putting Ned in charge. Very little damage control needed. But if half the nobles in Kings Landing hear what they've long suspected anyway only to have Tywin say it's a lie, his credibility is forever shot. The Mountain would have no reason to lie in that situation and everybody knows that. Which I suppose is why Oberyn chose to do it there in the first place. Sure maybe Tywin is allowed to take the black, but that would require him confessing, which I doubt his ego allows him to do. Maybe they could do another trail by combat but after Oberyn obliterated the most dangerous man in Westeros who would even attempt to fight him. Especially given how it's a convenient opportunity to cripple the Lannisters. I don't think anyone worthwhile steps up to fight for Tywin, and even if they do, my money's on Oberyn. Technically speaking the Mountain won that fight, but anyone who actually saw it saw him get his ass kicked. The fight wasn't close. Oberyn basically killed himself by letting his guard down. If he hadn't done that, and I watched that fight I'd keep my distance. Tywin would be a goner. Again you're right he could take the black but it would require HIM putting his ego aside for the sake of the family for once. And he doesn't have it in him,

[Mithras]

Martells already had a just cause to kill him. They knew that it was Tywin's work.

[NOT A TARG I SWEAR]

^ This. Varys is making big moves. The Old Lion has to go... no way can Varys get his mummer's dragon into place with Tywin Lannister as Hand.

Part of the Kevan killing is due to the fact that Aegon decided to make an early Westerosi appearance which was largely helped put into motion by Tyrion during their meeting. If Tyrion isn't on the lam after having escaped prison for killing the king and then killing his father, this meeting likely doesn't occur and Varys isn't forced to kill Kevan when he does in the manner in which he does it.

Maybe Varys would do this eventually, but it probably wouldn't have occurred yet in the novels and I think it would likely be done by other means. I think this was a last second improvise by Varys as his hand was forced by Aegon's audible.

[White Ravens]

If Oberyn had won his fight with fight with the mountain he would have pulled the spear from Gregor's body and hurled it at Tywin but the spear would be blocked before it got there by Bran warging Hodor. Bran had anticipated this event by tuning in to the weirnet and he wanted Tywin left alive, but his efforts were wasted because Bloodraven has also been tuning in to the weirnet for much longer so he wargs a giant who knocks Brodor aside and kills Tywin with his bare gigantic hands.

[Aemond's Eye]

What if they got a confession from Gregor? Would it be any better than Robert's will, which was torn apart by Cersei? Tywin would refuse it and there would be another trial by combat at best with an uncertain end. Even if Tywin was found guilty in a trial, he would be allowed to take the black and House Lannister would continue working. Tywin's guilt does not bind his descendants.

One should also note that even if it was revealed that Tywin gave the orders, he could easily say "So what?".

There is no written law that during a sack, women and babies should be spared, even if they are of noble blood.

That's because the rape and murder is illegal anyway - it's just an accepted part of warfare, as a 'reward' for the victorious troops. It's not exactly a case of Elia's rape and murder and the brutal murders of her children weren't a crime, so much as there's nothing to prove that Tywin is behind it. Tywin knows it's a crime, as he had hoped to just pin it all on Lorch after his death.

[oldworldtech]

Tywin was very smart and he called Oberyn at KL because he needed as many allies as possible against Stannis. Despite being humiliated, he wouldn't open an another war.

What would have happened though is that he would have send Oberyn to Dorne (because otherwise Obi would have killed him) and Tyrion would have gone with Oberyn (because otherwise Cersei would have assasinated him). And then obviously, Varys would have killed Tywin (similarity to how he killed Kevan) because Aegon would struggle more to come to throne with Tywin there, and because he hates Lannisters.

Tywin called Prince Doran not Prince Oberyn to KL. Doran sent Oberyn to piss off Tywin and to stir up drama. I do agree that Varys would have dispatched Tywin. Tyrion likely would have gone to Dorne and gotten involved in their plots. Although I expect that Tyrion would be more interested in making sure Myrcella was cared for and not mistreated as opposed to crowning her. After seeing what has been happening to kings he would be loathe to crown his niece.