I still don't have any proof, but if Shae was working for Tywin, then she was planted by Bronn... thus Bronn is Tywin's agent. Therefore, in addition to Tyrion commanding Bronn to keep him alive in that battle, Bronn was under orders from Tywin to keep Tyrion alive. (Or conversely, to not help Tyrion. Thus removing Tywin a huge embarassment at no cost or blame to himself.) Plus, I think Tywin constantly pushed Tyrion to see what he was capable of... I know that Tywin despised Tyrion, but he saw that Tyrion was a thinker and a planner. I believe Tywin harrassed Tyrion to bring forth a powerful intellect to assist House Lannister... the Lannisters already abound in money, swords, and power and Tywin's other children seem to not possess any desire to plan and scheme effectively on the grand level. Oh, Cersei thinks she can plot to secure the throne, but we all know her inadequacies. And Jaime only desires to be a swordsman... the greatest swordsman, but still only a man with a sword. Leaving Tyrion to use his brain to keep his older siblings in power, wealth, and swords.
Now as to the crossbow shot... It was murder. It was patricide. It was treason. It was not very noble to shoot an unarmed man on the pot. But could Tyrion claim self-defense?
Tywin denied Tyrion his birthright, Casterly Rock. Tywin spied on Tyrion. Tywin ordered Tysha to be raped by his barracks. Tywin ordered Tyrion's marriage annulled. Tywin left Tyrion exposed in the battle. Tywin repeatedly threatened Tyrion with emotional distress, harming Tyrion's women, and physical abuse. And Tywin showed no hint of fear... Tyrion was absolutely right to fear Tywin, even on the pot.
But in the end, it was murder.
But that does not mean that Tywin should not have seen it coming. How much abuse and disuse could Tyrion take before he snapped? Of course, Tyrion was safely locked away until his scheduled execution in a few hours...
How much remorse will Tyrion feel? He's now killed both of his parents.