feat(api): add Better Auth plugin and protected GET /me endpoint

- authPlugin runs preHandler that turns Fastify request headers into a Headers
  object, calls auth.api.getSession, and decorates request.user / request.session
- requireAuth() preHandler short-circuits with 401 when not signed in
- New module modules/me with GET /me returning the authenticated user/session
- env validates BETTER_AUTH_SECRET (≥32) and BETTER_AUTH_URL — must match web
- Restored .js extensions in shared packages so NodeNext-resolution consumers
  (api) typecheck cleanly; Next webpack now uses extensionAlias to map .js → .ts
- Re-enabled NodeNext for packages/auth and packages/db tsconfigs

Verified end-to-end:
- POST /api/auth/sign-in/email on web returns session cookie
- GET /me on api with the cookie returns 200 + user/session
- GET /me without the cookie returns 401

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

apps/api/package.json

@@ -15,8 +15,10 @@
     "@fastify/helmet": "^11.1.1",
     "@fastify/rate-limit": "^9.1.0",
     "@fastify/sensible": "^5.6.0",
+    "@ketopath/auth": "workspace:*",
     "@ketopath/db": "workspace:*",
     "@ketopath/shared": "workspace:*",
+    "better-auth": "^1.0.21",
     "fastify": "^4.28.1",
     "fastify-plugin": "^4.5.1",
     "zod": "^3.23.8"