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/.oxlintrc.json

@@ -0,0 +1,14 @@
+{
+  "$schema": "./node_modules/oxlint/configuration_schema.json",
+  "extends": ["../.oxlintrc.json"],
+  "plugins": ["eslint", "typescript", "unicorn", "oxc", "react", "nextjs", "import"],
+  "categories": {
+    "correctness": "error",
+    "suspicious": "warn"
+  },
+  "rules": {
+    "react/react-in-jsx-scope": "off",
+    "import/no-unassigned-import": "off"
+  },
+  "ignorePatterns": ["node_modules", ".next", "dist", "build", ".source", "public/openapi.json"]
+}