It was obvious Joffrey was lying. What butcher's boy in his right mind would attack the heir to the Seven Kingdoms? Even Sansa's "I don't remember" version of "no comment", was a tip off. Robert's response was to do nothing, except to have Sansa's direwolf killed.
Robert actually did respond though, he looks at Joffery and states about how he let himself be disarmed by a little girl. So Robert is openly admitting that Joffery is lying.
But despite that, he still wants Nymeria dead and still orders the death of Lady. And we know that Micah was already dead anyway, so in the end it didn't matter that Joffery was lying and that Robert admitted to it, it was Micah, Arya and Sansa who were punished.
And someone else can do those things? Tyrion was in a good position to at least help Sansa get on with her life. She could have had all the books, music, art, or training in swordsmanship or any other subject she wanted. She could have travelled the world. He would have built her a house or a castle anywhere she wanted, or sent her anywhere she wanted to go, if she asked. And although he could not kill Cersei, Joffrey, or Tywin without becoming a kinslayer, he would have given her a big enough allowance that she could have hired her own assassins, and probably wouldn't have asked her about it. As a Lannister, Sansa could afford to hire anybody in the world to do just about anything. Too bad it didn't occur to her.
The problem with this sort of reasoning is you have to assume that Sansa really had the ability to just ask to leave or get what she wanted. It wasn't Tyrion's money for example, it was his family's. Would Tywin really allow Sansa to leave King's Landing? I don't believe so. Sansa was still alive because Tywin wanted Stark heir's to Winterfell. I feel that once she had a son or two, her death warrant would have been sealed. I think maybe on some level she knew that too.
And I mean, when she asks Tyrion if they could leave the reception, she isn't even granted that, because Joffery refuses. If Tyrion didn't even have the power to be able to leave that reception with Sansa, I am not sure we can believe that Sansa was ever going to be allowed to leave King's Landing, or that Tyrion had any power or say so in letting her leave.
Personally, I don't think that Tyrion's inactivity means that he is siding with his family. I mean, what can Tyrion do knowing that Joffrey isn't the rightful king? Or, if they had lost the war, what would have happened to Tyrion? This is not saying that I'm necessarily concerned for him, but how many people in series care that Tyrion hasn't done anything to harm anyone or doesn't agree with his family--very very few. All that matters to them is that he is a Lannister. So, yes, he wants his family to win the war because the little protection he had is gone if they lose. Can Tyrion defy his father? He can crack smartass remarks, but all of his security is because of his father. Without that, he life looks pretty grim. Regardless of what Tyrion feel, his family is going to do what they want to do.
I do think Tyrion is siding with his family to some degree. He does love Jamie and wanted him to come home safely. So much of what he did was to ensure that. He also of course was looking out for his own skin at the Blackwater battle. He knew his head would be on a pike whether he helped or not.
However, I don't think from Sansa's standpoint we could assume she would look at these actions and think of Tyrion as being blameless. I think it's logical that she would or could see him as being the enemy, even if he truly wasn't. And there's also the matter of Tyrion still being arrested by her mother, for the attempt on Bran's life. We've never heard what she thinks about that, or if she even knows. But if she does, then that would be another thing she might hold against him.
The death of Lady represented the end of Sansa's Stark-ness, which she had already lost when she prevaricated about what happened on King's Road. A true Stark would have blundered on ahead, and told the true story of what had happened that day, come what may.
I disagree that Sansa had no choice. Robert was king, not Joffrey, and he would have attempted to discipline Joffrey. Sansa sacrificed Arya's trust, Mica's life, Lady's life, and Nymeria's life, for her relationship with Joffrey, so in a way it's justice that that relationship turned into a nightmare. Sansa has paid in full, though, and I completely forgive her.
I don't agree with this at all. She didn't sacrifice Micah's life because he was already dead by the time this happened. She didn't sacrifice Nymeria's life, because even after Robert admitted that Joffery was lying and that he believed Arya and Micah had no wrong doing, he was willing and ordering Nymeria be killed.
And finally she is responsible for Lady's death, but not for the reason that many believe. If Sansa had said that Joffery was telling the truth, it is much more likely that Lady would have been spared(at least for that moment, as I am sure Cersei would have found a way to kill Lady eventually). However, despite Robert calling Joffery a liar in open court, Robert still orders Lady's death.
It looks to me that Sansa's statement was simply the only way to prove that Joffery was lying, without calling him a liar. And we know that she couldn't do that, from Ned.
It simply comes down to this statement:
Robert was king, not Joffrey, and he would have attempted to discipline Joffrey.
He doesn't. He never punished Joffery at all, even after admitting that Joffery was the liar.
Whether you believe Sansa could have called Joffery a liar or not, it doesn't matter(and I don't think she could). The end result was still that Joffery was proven to be a liar and nothing happened to him. It was still everyone else who was punished.
And while, I don't believe that Robert would have burned Arya alive or had Sansa killed if she had also called Joffery a liar. We see the results of Robert agreeing that Joffery was a liar. So if that was Arya being proved innocent, then how much worse could it have been if she was decreed to be at fault?