feat(biome.json): replace ESLint and Prettier with Biome

[TylerAdamMartinez]

Why

This PR addresses the following problem/context:

• Replace the existing ESLint and Prettier toolchain with Biome to simplify linting and formatting.
• Standardize the development experience by using a single tool for formatting, linting, and import organization.
• Reduce project configuration and maintenance overhead.

How

Implementation summary - the following was changed/added/removed:

• Added a biome.json configuration with project linting and formatting rules.
• Removed the existing ESLint configuration (eslint.config.ts) and Prettier configuration (.prettierrc).
• Added Biome as a development dependency and removed ESLint- and Prettier-related dependencies.
• Updated VS Code workspace settings and recommended extensions to use the Biome extension and enable Biome code actions on save.
• Updated linting documentation (LINTING.md) to reflect the new Biome workflow and commands.
• Updated project dependencies and lockfile to reflect the migration.

Notes

Any special considerations, workarounds, or follow-up work to note?

• The initial Biome configuration preserves several lint rules as warnings to allow incremental cleanup rather than blocking development. @chasetmartin

[github-actions]

Preview Deployment

Preview URL: https://preview-task-update-linting-toolchain-auejgdbofq-uc.a.run.app

Note: This preview uses the staging API endpoints.

[chasetmartin]

Nice, looks like a cool and widely adopted tool (you love Rust don't you 😎). I don't see an issue with this from my end, looks like the linting in CI with Biome is working as expected and it is extremely fast when run locally.