chore(tooling): tweak cache directories and switch to ty

[mridang]

Description

This PR migrates the project's type checking tooling from mypy to ty (Astral's new Rust-based type checker). The migration includes updating the GitHub Actions workflow, devbox configuration, and pyproject.toml dependencies.

Related Issue

N/A - Tooling modernization

Motivation and Context

ty is an extremely fast Python type checker written in Rust by Astral (the creators of ruff). Migrating from mypy to ty provides significantly faster type checking performance and aligns our tooling stack with modern Rust-based Python tools.

How Has This Been Tested?

• Ran poetry run ty check locally to verify type checking works correctly
• Verified the GitHub Actions workflow executes successfully
• Confirmed all existing type annotations are compatible with ty

Documentation:

N/A - Internal tooling change

Checklist:

• I have updated the documentation accordingly.
• I have assigned the correct milestone or created one if non-existent.
• I have correctly labeled this pull request.
• I have linked the corresponding issue in this description.
• I have requested a review from at least 2 reviewers
• I have checked the base branch of this pull request
• I have checked my code for any possible security vulnerabilities