What was the core of Tywin and Tyrion's conflict?

Tywin Lannister held Tyrion responsible for his mother's death (Joanna Lannister died giving birth to Tyrion) and despised his dwarfism as a humiliation to Lannister pride. Tyrion was brilliant, loyal to the family, and capable of solving problems none of his siblings could manage — and Tywin knew it, used it, and never acknowledged it. This is the wound that runs through everything.

Did Tywin ever respect Tyrion?

Functionally, yes. He appointed Tyrion acting Hand when Joffrey was unsuitable. He used Tyrion to negotiate, strategize, and manage difficult political situations. He deployed his intelligence constantly while refusing to credit it publicly. The respect was operational. It was never offered as love or recognition.

What was Tyrion's strongest desire in this relationship?

Simple acknowledgment. He wanted to hear from his father that his existence wasn't a catastrophe. He never got it. Even Tywin's final conversation with Tyrion — confronted in the privy, crossbow aimed at him — Tywin calls Shae a whore. He couldn't offer his dying son the one thing the son needed.

How did Tyrion killing Tywin change him?

It freed him in the immediate term and hollowed him in the long run. He'd fantasized about escaping Tywin's judgment. The reality was that Tywin was also his structure — the thing he oriented himself against. Without Tywin, Tyrion's purpose becomes murkier. His seasons in Essos reflect this disorientation.

What does this relationship teach about toxic parental legacies?

That the most damaging thing a parent can do to a capable child is withhold recognition. Tyrion would have done almost anything for Tywin. The tragedy isn't Tyrion's rebellion — it's how long it took him to stop trying to earn what was never going to come.