Did Tywin know why Jaime killed Aerys Targaryen? And how did he feel about the event?

Question

As we all know, Aerys Targaryen commanded Jaime Lannister to kill his own father, Tywin Lannister. To which, Jaime refused and killed Aerys instead.

Did Tywin ever know about Aerys' command?

How did Tywin feel about Jaime killing his king? Tywin Lannister was one to use underhanded tactics when they were necessary, but would he go so far as to honor someone killing their king when they were fully sworn to them under a life-long oath? If Tyrion weren't stunted, or if he had another son, would he still treat Jaime the same, post-Kingslayer or would he begin to resent him for his actions?

Answer 1

tl;dr: No one knew about the command¹, until Jaime told Brienne.

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Jaime the secret saviour

Jaime has never told anyone about the reason he killed Aerys, not even his own family. You would have thought that he'd tell Cersei, but he hadn't.

Everyone knows him as "Kingslayer", and it's because they think that he betrayed his vows².

I think it passing odd that I am loved by one for a kindness I never did, and reviled by so many for my finest act.
^{-A Song of Ice and Fire: Book Two - A Clash of Kings, Chapter Fifty-Five (Catelyn VII).}

...is the first time that he has made mention in any way to this.

Jaime eventually tells, Brienne fully (in the bath scene at Harrenhall), setting off his arc of redemption (sort of), and she's the first person he's ever confided in.

“It fell to me to hold the Red Keep, but I knew we were lost. I sent to Aerys asking his leave to make terms. My man came back with a royal command. ‘Bring me your father’s head, if you are no traitor.’ Aerys would have no yielding. Lord Rossart was with him, my messenger said. I knew what that meant.
“When I came on Rossart, he was dressed as a common man-at-arms, hurrying to a postern gate. I slew him first. Then I slew Aerys, before he could find someone else to carry his message to the pyromancers. Days later, I hunted down the others and slew them as well. Belis offered me gold, and Garigus wept for mercy. Well, a sword’s more merciful than fire, but I don’t think Garigus much appreciated the kindness I showed him.”
^{-A Song of Ice and Fire: Book Three - A Storm of Swords, Chapter Thirty-Seven (Jaime V).}

...

“Has my tale turned you speechless? Come, curse me or kiss me or call me a liar. Something.”
“If this is true, how is it no one knows?”
“The knights of the Kingsguard are sworn to keep the king’s secrets. Would you have me break my oath?” Jaime laughed. “Do you think the noble Lord of Winterfell wanted to hear my feeble explanations? Such an honorable man. He only had to look at me to judge me guilty.” Jaime lurched to his feet, the water running cold down his chest. “By what right does the wolf judge the lion? By what right?”
^-Ibid.

People of Westeros are bound by the codes of honour, Jaime had slewn his king, the king he had vowed to protect, no matter the reason, people were ready to dismiss him before hearing what he had to say.

Tywin the blind

Tywin don't give a rat's, because Jaime is the Golden Boy of the family, Tywin will always support Jaime, not matter what. He even ensured that Jaime wasn't prosecuted, rather pardoned by Robert Baratheon.

You're right to question Tywin's motives had he had another son to choose from. Tywin was extremely displeased at Jaime becoming a Kingsguard, had Tywin had another choice, he certainly would have forsaken Jaime.

Tywin's allegiance is to his family, above all. So the thought of a viable strong heir, instead of Jaime being in the Kingsguard would have become his priority.

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^{1. Except a very small number of unconfirmed exceptions; such as Varys who knows everything, and an unknown messenger boy.}

^{2. See: Why do people of the realm hate Jaime Lannister?}