Migrate from ESLint/Prettier to Biome for linting and formatting

[trillium]

Overview

Replace ESLint and Prettier with Biome - a single, faster tool that handles both linting and formatting with built-in auto-fix.

Action Items

• Remove ESLint packages and config (eslint.config.mjs)
• Remove Prettier packages and configs (.prettierrc, .prettierignore)
• Add @biomejs/biome to root devDependencies
• Create biome.json config with equivalent settings
• Update npm scripts (lint, format, check)
• Update .lintstagedrc.json for pre-commit hooks
• Remove ESLint/Prettier deps from backend and client package.json

Resources/Instructions

• Biome Documentation
• Migrating from ESLint/Prettier

Related Issues

This could close or supersede #1966, #1967, #1969