Did Varys set Tyrion up?

Question

In A Storm of Swords,

Varys releases Tyrion from prison, and Tyrion takes the opportunity to kill Lord Tywin.

But a few things seem off at the surface.

Tywin supposedly hates whores, but apparently has taken Shae to bed.

Also

Shae doesn't seem at all surprised by seeing Tyrion.

This implies to me that Varys may have manipulated things to get Tyrion into a certain emotional state to become his unwitting tool. Granted, it wouldn't be a sure thing but Varys always seems to have a few irons in the fire.

Is there any evidence beyond what I've said to support this?

Answer 1

It's unclear if Varys indeed set Tyrion up to go to the Tower of the Hand, however, I will challenge a few things with your assumptions for why Varys set him up.

Tywin may not have hated whores

As explained in this answer to a related question it would appear that Tywin didn't have anything against whores only being publicly seen with them.

You will not take the whore to court, Lord Tywin had commanded.

_{A Clash of Kings, Tyrion I}

"As was your mother." Lord Tywin rose abruptly, to tower over his dwarf son. "Go back to your bed, Tyrion, and speak to me no more of your rights to Casterly Rock. You shall have your reward, but it shall be one I deem appropriate to your service and station. And make no mistake—this was the last time I will suffer you to bring shame onto House Lannister. You are done with whores. The next one I find in your bed, I'll hang."

_{A Storm of Swords, Tyrion I}

Basically Tywin is saying if I ever find/find out about a whore with you I will not be impressed but there's no objection to him just visiting one. In fact there's even speculation that Tywin was the one who had the tunnel dug to Chataya's.

"How is it a brothel happens to have a secret entrance?"

"The tunnel was dug for another King's Hand, whose honor would not allow him to enter such a house openly. Chataya has closely guarded the knowledge of its existence."

_{A Clash of Kings, Tyrion III}

George R. R. Martin has also said that it is an "interesting theory" that Tywin was the Hand who had this tunnel built.

[Was Tywin the Hand who made the secret tunnel into Chataya's, to hide his visits?]

GRRM: Interesting theory.

_{So Spake Martin, Asshai.com Forum Chat}

Shae did appear to be surprised at finding Tyrion

Her reaction is one very much at surprise, though her not knowing Varys might plan to send Tyrion doesn't mean it wasn't Varys' intention.

"M'lord?" a woman's voice called.

That might have hurt me once, when I still felt pain. The first step was the hardest. When he reached the bed Tyrion pulled the draperies aside and there she was, turning toward him with a sleepy smile on her lips. It died when she saw him. She pulled the blankets up to her chin, as if that would protect her.
"Were you expecting someone taller, sweetling?"

_{A Storm of Swords, Tyrion XI}

Varys protested?

Now as for if Varys had planned for Tyrion to go there. It could be a case that Varys protested too much for Tyrion going there whilst also giving detailed instructions for how to get to Tywin's chambers in the tunnels. It seems to be left open to interpretation if Varys was really trying to stop Tyrion or if that's what he wanted.

"Yes." Frozen hinges screamed in protest as Varys pulled open a long-closed door. Flakes of rust drifted to the floor. "This will take us out to the river."
Tyrion walked slowly to the ladder, ran his hand across the lowest rung. "This will take me up to my bedchamber."

"Your lord father's bedchamber now."

He looked up the shaft. "How far must I climb?"

"My lord, you are too weak for such follies, and there is besides no time. We must go."

"I have business above. How far?"

"Two hundred and thirty rungs, but whatever you intend—"

"Two hundred and thirty rungs, and then?"

"The tunnel to the left, but hear me—"

"How far along to the bedchamber?" Tyrion lifted a foot to the lowest rung of the ladder.

"No more than sixty feet. Keep one hand on the wall as you go. You will feel the doors. The bedchamber is the third." He sighed. "This is folly, my lord. Your brother has given you your life back. Would you cast it away, and mine with it?"
"Varys, the only thing I value less than my life just now is yours. Wait for me here." He turned his back on the eunuch and began to climb, counting silently as he went.

_{A Storm of Swords, Tyrion XI}

The room was prepared

Now whilst not being proof it does seem lucky that Tywin's dagger was left on the bedside table and the crossbow had a box directly underneath it so Tyrion could reach it.

Afterward he found Lord Tywin's dagger on the bedside table and shoved it through his belt. A lion-headed mace, a poleaxe, and a crossbow had been hung on the walls. The poleaxe would be clumsy to wield inside a castle, and the mace was too high to reach, but a large wood-and-iron chest had been placed against the wall directly under the crossbow. He climbed up, pulled down the bow and a leather quiver packed with quarrels, jammed a foot into the stirrup, and pushed down until the bowstring cocked. Then he slipped a bolt into the notch.

_{A Storm of Swords, Tyrion XI}

Jaime was the catalyst

As is pointed out in @JAD's answer Jaime telling Tyrion the story of how Tysha wasn't actually a whore seems to be what pushed Tyrion over the edge.

"Tysha?" His stomach tightened. "What of her?"

"She was no whore. I never bought her for you. That was a lie that Father commanded me to tell. Tysha was... she was what she seemed to be. A crofter's daughter, chance met on the road."

[...]

"Oh, you've earned more than that, Jaime. You and my sweet sister and our loving father, yes, I can't begin to tell you what you've earned. But you'll have it, that I swear to you. A Lannister always pays his debts." Tyrion waddled away, almost stumbling over the turnkey again in his haste. Before he had gone a dozen yards, he bumped up against an iron gate that closed the passage. Oh, gods. It was all he could do not to scream.

_{A Storm of Swords, Tyrion XI}

And as is pointed out Jaime certainly didn't want to his father dead nor did he know where Varys was so it seems unlikely he had any part in the plan.

He never said he meant to kill our father. If he had, I would have stopped him. Then I would be the kinslayer, not him.

Jaime wondered where Varys was hiding. Wisely, the master of whisperers had not returned to his own chambers, nor had a search of the Red Keep turned him up. It might be that the eunuch had taken ship with Tyrion, rather than remain to answer awkward questions. If so, the two of them were well out to sea by now, sharing a flagon of Arbor gold in the cabin of a galley.

_{A Feast for Crows, Jaime I}

Varys the opportunist

It could just be that Varys didn't plan any of this but seeing the opportunity for someone to take out a great military leader of the enemy, he took it. This could seem likely as he would undoubtedly not have prepped Jaime for telling Tyrion about Tysha.

---

In conclusion there is nothing pointing either way for if Varys wanted Tyrion to visit his father or not but there are a few signposts that could look like he wanted him to.

Answer 2

It's debatable. There are some theories that Tywin's room was prepped so that Tyrion could reach the crossbow:

Afterward he found Lord Tywin's dagger on the bedside table and shoved it through his belt. A lion-headed mace, a poleaxe, and a crossbow had been hung on the walls. The poleaxe would be clumsy to wield inside a castle, and the mace was too high to reach, but a large wood-and-iron chest had been placed against the wall directly under the crossbow. He climbed up, pulled down the bow and a leather quiver packed with quarrels, jammed a foot into the stirrup, and pushed down until the bowstring cocked. Then he slipped a bolt into the notch.
_{A Storm of Swords, Tyrion XI, emphasis mine}

However, I don't think that from the moment Jaime freed Tyrion from the cell, it was entirely given that Tyrion would actually go up into the Tower of the Hand.

As outlined in this question, the biggest reason for Tyrion to go up there is the confession by Jaime.

Jaime confesses that the story he told about how Tysha (Tyrion's former wife) was a whore, was actually a lie he was forced to tell by Tywin. This was very likely the final straw, given that he asks Tywin at boltpoint about her repeatedly.

"Tysha?" His stomach tightened. "What of her?"

"She was no whore. I never bought her for you. That was a lie that Father commanded me to tell. Tysha was . . . she was what she seemed to be. A crofter's daughter, chance met on the road."

Tyrion could hear the faint sound of his own breath whistling hollowly through the scar of his nose. Jaime could not meet his eyes. Tysha. He tried to remember what she had looked like. A girl, she was only a girl, no older than Sansa. "My wife," he croaked. "She wed me."

"For your gold, Father said. She was lowborn, you were a Lannister of Casterly Rock. All she wanted was the gold, which made her no different from a whore, so . . . so it would not be a lie, not truly, and . . . he said that you required a sharp lesson. That you would learn from it, and thank me later . . ."

"Thank you?" Tyrion's voice was choked. "He gave her to his guards. A barracks full of guards. He made me . . . watch." Aye, and more than watch. I took her too . . . my wife . . .

"I never knew he would do that. You must believe me."

"Oh, must I?" Tyrion snarled. "Why should I believe you about anything, ever? She was my wife!"

“Tyrion –it...”

He hit him. It was a slap, backhanded, but he put all his strength into it, all his fear, all his rage, all his pain. Jaime was squatting, unbalanced. The blow sent him tumbling backward to the floor. “I... I suppose I earned that.”

"Oh, you've earned more than that, Jaime. You and my sweet sister and our loving father, yes, I can't begin to tell you what you've earned. But you'll have it, that I swear to you. A Lannister always pays his debts." Tyrion waddled away, almost stumbling over the turnkey again in his haste. Before he had gone a dozen yards, he bumped up against an iron gate that closed the passage. Oh, gods. It was all he could do not to scream.
_ibid