What is Kakashi's core teaching about failure?

That failure is data, not verdict. He doesn't tell Team 7 that failure is okay — he teaches them why it happened and what to do differently. The bell test is a controlled failure: Team 7 fails, gets no lunch, and learns that the lesson isn't about fighting ability at all. It's about working together.

How did Kakashi himself fail?

Multiple times, badly. He failed to save Obito (or thought he did). He failed to save Rin. He failed to prevent his father's death. Each failure became part of his teaching methodology. He doesn't lecture about failure because he knows what it actually costs.

What does Kakashi think about talent versus effort?

He respects both, but he respects intelligent effort most. His genius students (Sasuke, Sakura at her peak) need to learn to channel talent; his less naturally gifted students (early Naruto) need to learn to find angles. Both need failure to teach them where their real limits are. Kakashi lets them hit those limits before intervening.

What does Kakashi model about carrying grief while continuing to function?

Everything. He visits the memorial stone regularly. He's chronically late because he's talking to Obito and Rin's names carved in stone. He's visibly, persistently sad — and he also shows up, teaches, leads, and protects. He doesn't perform healing. He carries grief and does the job anyway, and that combination is the most realistic model of resilience in the series.

What is the lesson Kakashi gives Team 7 that lasts longest?

"In the ninja world, those who abandon their comrades are worse than trash." Not as a rule — as a value. There's a difference. Rules can be broken for good reasons. Values are what you carry into situations where no rule applies. He teaches them to carry that value into the unknown.