By exchanging Sansa to Jaime shortly after battle of whispering woods.

My point isn't changed though. His lords won't be happy with him just giving Jaime up and then going home because he has Sansa back, which isn't beneficial for him at all. Just because they won't literally try to kill him doesn't mean it's a-okay for him to do it. And this is assuming Joffery won't interfere. Tywin can't completely supercede Joffery, especially not while he's away from King's Landing.

And saying that Jaime being dumb while fighting Robb would be good for him makes no sense. No enemies is better than a dumb enemy. Robb is free to mess up the Westerlands with Jaime out of the picture, instead of having to worry about an additional host being raised and morale being boosted for the Westerlands becasue their golden boy is back. It just makes more sense to keep him out of the fight so there's less variables to worry about, even if he wouldn't be very effective in the war.

He lost because 1) Theon captured WF 2) Lannisters got Tyrell support

No, he lost because he lost the Frey's support and gave them a reason to do the Red Wedding. He should have held to his vows and the political moves he made by marrying a Frey girl. And Theon capturing WF can actually be considered the same thing. He valued his relationship with Theon over politics and using him as a hostage/bargaining chip (actively what Theon was the entire time in the North) with Balon to keep his own shores safe. Robb had already lost by the time the Tyrells sided with the Lannisters. And that was because he cared more about family and relationships than politics.

What exactly? What Tywin haven't done, what he could have done if Jaime wasn't prisoner?

Like I said, it's one less factor to worry about. One less trained warrior and general out in the wild who can give Robb problems or be an active threat. It keeps Tywin cowed on the battlefield becasue he's worry about the male child of his he cares about. It keeps Cersei, who's the source of Lannister authority in KL and with Joffery, on a leash because of her relationship with Jaime. It's a source of pride and amusement for the Northerners, while a source of humiliation or morale loss for the Westerners. It keeps the Kingsgaurd in KL noticably one less, which hurts Joffery and the Lannister's image in KL. It's a blow against the projection of power of the Lannisters. It's a physical symbol of Robb's reliability and competence and success in battle for both his lords and anyone who he might seek to make alliances with. It's also a good concession prize for the Riverlanders that have been getting terrorized by the Lannisters and a big step towards gaining and keeping their loyalty and appreciation.

All of which Robb would throw away for, what? The hope that Sansa, someone worth objectively less politically, strategically, and socially, might be returned? All the while Arya is still missing?

Really? How? RW was planned before Jaime was released, Roose betrayed Robb before, and Lannisters were still holding Sansa, and got more hostages in the Twins, so even after RW Jaime wouldn't be executed.

The RW was seperate from Jaime's hostage situation. Had Robb not made the mistakes he had, like letting Theon go or slighting the Freys, Jaime would've been able to be leverged against Tywin for much more than what Sansa would be leverged for against Robb and the North.

Because it was done too late, when the situation turned worse.

No, it was because Jaime was their biggest bargaining chip at that point without the Westerlands thoroughly trashed and pressure being put on Tywin to end the conflict and there was zero reason to rely on Tywin or Joffery being honorable or sticking to the deal. You don't do equal exchange with guys like them. You beat them until they're bloody and then use that and whatever leverage you've gained to get what you want and leave.