[DevTask]: Replace lint/typecheck/format in Claude.md with Claude hooks

This issue proposes a move away from prodding Claude with Claude.md to run commands, and instead simply running them with a hook.

Our Claude.md currently states:

```md
...(3) run typecheck, lint, format (with prettier)...
```

Instead of guiding Claude to do these tasks, we can instead add a Claude Code hook to run on stop that will execute our linting commands.

https://docs.anthropic.com/en/docs/claude-code/hooks#stop

This can be stored in the project's `.claude/settings.json`.

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