Why Littlefinger bothered to kill Jon Arryn and assassinate Bran Stark?
Jon Arryn have already known too much about Lannisters' incest, so I would expect that he will be killed by Lannisters or someone else for sure.
Also why bother to kill Bran Stark and frame Lannisters when Lannisters would already have motive to kill Bran Stark and silence him for good? Wasn't Littlefinger's plot wasted to do it all by himself when it could be resolved in a way he wanted without his involvement anyway?
Littlefinger thrives on chaos.
Littlefinger explains his motivations for everything he does to Sansa:
Whenever I consider a question, I ask myself ‘Will this action make this picture a reality,’ pull it out of my mind and into the world… and I only act if the answer is yes. A picture of me on the Iron Throne and you by my side.
Littlefinger tricked Lysa Arryn into poisoning Jon Arryn. He then sent a letter to Catelyn Stark, accusing the Lannisters. His goal here is clearly to get the Lannisters to fight the Starks. The same is true with the assassination attempt of Bran - he plants the idea in Catelyn's head that Tyrion is responsible. This leads directly to all the chaos that follows. And we know what Littlefinger thinks of chaos -
Chaos is a ladder. Many who try to climb it fail and never get to try again. The fall breaks them. And some, are given a chance to climb. They refuse, they cling to the realm or the gods or love. Illusions. Only the ladder is real. The climb is all there is.
Littlefinger had hoped to climb the ladder of chaos until he reached the top.
Littlefinger did need to be the one to get the wheels turning on this plot.
There is no evidence that Cersei were aware of Jon Arryn's investigation into her children. If Jon found enough proof then when to King Robert and was able to convince him of the treachery, Cersei would have been arrested on the spot. The reason Ned's plan fails later is that Robert is wounded on the boar hunt and Ned warns Cersei directly of his intents.
Littlefinger needed the plot to have the Starks and Lannisters at each other.
We know Littlefinger ultimate motive is "the Iron Throne with you [Sansa] by my side. To make this a possibility you need to have two things, 1) the Iron Throne be in contention, and 2) Sansa available for marriage. The Lannisters stand in the way of the throne and Starks (Ned in particular) stand in the way of Sansa. This discord is sowed by the letter Lysa writes to Caetlyn, and by falsely linking Tyrion to the assassination attempt of Bran. The attempt on Bran was not in his initial plan, but he seized the opportunity from the chaos.
But why Tyrion?
I think the reason Littlefinger chose to blame Tyrion is that he (Tyrion) was not in the capital at that time and was therefore vulnerable to arrest. It would have been nigh impossible to arrest a high ranking Lannister within a city they controlled. With Tyrion being in the North, it would have been a better chance to to find and arrest him there with the support of the other northern lords.
It makes no sense. There's also no support for the accusation. In the books it's deduced but never proven that it was Joffrey who sent the assassin.
It fits Joffrey much better because it ties in with his expressed sentiment that it would be more merciful to kill Bran than let him live as a cripple.
In the end I think the Stark Sisters just use it as more (albeit false) evidence that they didn't really need.
There is always the possibility that Littlefinger orchestrated the death of Jon Arryn, and the assassination attempt on Bran at the request of the Lannisters.
As Master of Coin, with a large spy network of shady characters, it wouldn't be very odd for the Queen to reach out with a request.
Another possibility would be he murdered Jon Arryn to win Cersei's trust after finding out what Jons plan were, which in turn got him the 'contract' on Bran from Cersei.
The oddity is the dagger, who was it that wanted to frame Tyrion exactly? Littlefinger had no reason to frame him at the time, however, Cersei hated her brother and is psychotic enough to aim to rid herself of his presence.
Littlefinger worked on the plots to kill Jon Arryn and kill Bran Stark because he was secretly working with/for the Lannisters in season 1 and in the year before season 1.
Cersei wanted Jon Arryn dead because she feared he would learn that her children were incest bastards of her brother, and not true heirs to the Iron Throne.
Cersei wanted Bran Stark dead because the boy saw Cersei committing incest with her brother, Jaime. If Bran mentioned it after recovering from the fall, King Robert Baratheon would have her killed for adultery. Robert would also kill Jaime and the children.
Cersei probably appealed to Littlefinger to help with the plots against Bran Stark and Jon Arryn.
Littlefinger likely went along with the plot to kill Jon Arryn because with Jon Arryn dead, Littlefinger could move forward on his plan to marry Lysa Arryn and become Lord Protector of the Vale.
By helping in the plot to kill Bran, he can implement the assassination plot in way that makes it look as if Tyrion Lannister was involved. That would sow distrust between the Starks and Lannisters, which would create chaos in the 7 kingdoms and allow him to advance another rung up his ladder.
We see a direct confirmation that Littlefinger worked for the Lannisters in season 1 when he betrayed Ned Stark. Ned Stark had learned Cersei's secret, and she could not allow Ned Stark to live. So she probably asked Littlefinger to betray him too.
It is only in a later season do we see Littlefinger working against the Lannisters. His early work for the Lannisters was just one rung on the ladder to his own personal success.
