refactor: replace Biome with oxlint + oxfmt

Migrate the entire monorepo from Biome 2.4.7 to oxlint 1.56.0 (linter)
and oxfmt 0.41.0 (formatter) for faster lint/format and broader rule
coverage.

- Add `.oxlintrc.json` with React, TypeScript, unicorn, import plugins
  and correctness/suspicious categories
- Add `.oxfmtrc.json` with 2-space indent, import sorting, and Tailwind
  class sorting (all 30+ custom className attributes migrated)
- Add `docs/.oxlintrc.json` and `docs/.oxfmtrc.json` with Next.js plugin
- Update all 12 workspace package.json scripts: `oxlint`, `oxfmt`,
  `oxfmt --check`
- Add `format:check` turbo task and CI step
- Update VS Code settings/extensions to use `oxc.oxc-vscode`
- Update CI path triggers from `biome.json` to new config files
- Remove all `biome-ignore` comments and fix shadowed variables
- Delete `biome.json` and `docs/biome.json`
- Reformat entire codebase with oxfmt

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

docs/package.json

@@ -6,8 +6,9 @@
     "build": "next build",
     "dev": "next dev",
     "start": "next start",
-    "lint": "biome check",
-    "format": "biome format --write",
+    "lint": "oxlint",
+    "format": "oxfmt",
+    "format:check": "oxfmt --check",
     "check-types": "fumadocs-mdx && next typegen && tsc --noEmit",
     "generate:api-docs": "bun scripts/generate-api-docs.ts",
     "postinstall": "fumadocs-mdx"
@@ -20,21 +21,23 @@
     "fumadocs-openapi": "^10.3.18",
     "fumadocs-ui": "^16.6.17",
     "lucide-react": "^0.577.0",
-    "next": "16.1.6",
+    "next": "16.1.7",
     "react": "^19.2.4",
     "react-dom": "^19.2.4",
     "shiki": "^4.0.2",
     "tailwind-merge": "^3.5.0"
   },
   "devDependencies": {
-    "@biomejs/biome": "^2.4.7",
-    "@tailwindcss/postcss": "^4.2.1",
+    "@tailwindcss/postcss": "^4.2.2",
     "@types/mdx": "^2.0.13",
     "@types/node": "^25.5.0",
     "@types/react": "^19.2.14",
     "@types/react-dom": "^19.2.3",
+    "oxfmt": "0.41.0",
+    "oxlint": "1.56.0",
     "postcss": "^8.5.8",
-    "tailwindcss": "^4.2.1",
+    "tailwindcss": "^4.2.2",
     "typescript": "^5.9.3"
-  }
+  },
+  "packageManager": "bun@1.3.10"
 }