Auth with passwords + Playwright E2E tests + PostHog analytics

- bcrypt password hashing in auth (register, login, change-password)
- Login/register pages with password fields
- Profile update + OAuth placeholder endpoints
- Playwright test suite: auth, pages, API (3 test files)
- PostHog Docker analytics on :8010

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

api/package.json

@@ -21,6 +21,7 @@
     "@fastify/swagger": "^9.7.0",
     "@fastify/swagger-ui": "^5.2.5",
     "bcrypt": "^6.0.0",
+    "bcryptjs": "^3.0.3",
     "dotenv": "^17.3.1",
     "fastify": "^5.8.4",
     "ioredis": "^5.10.1",