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-lock.json

@@ -17,6 +17,7 @@
         "@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",
@@ -578,6 +579,15 @@
         "node": ">= 18"
       }
     },
+    "node_modules/bcryptjs": {
+      "version": "3.0.3",
+      "resolved": "https://registry.npmjs.org/bcryptjs/-/bcryptjs-3.0.3.tgz",
+      "integrity": "sha512-GlF5wPWnSa/X5LKM1o0wz0suXIINz1iHRLvTS+sLyi7XPbe5ycmYI3DlZqVGZZtDgl4DmasFg7gOB3JYbphV5g==",
+      "license": "BSD-3-Clause",
+      "bin": {
+        "bcrypt": "bin/bcrypt"
+      }
+    },
     "node_modules/binary-extensions": {
       "version": "2.3.0",
       "resolved": "https://registry.npmjs.org/binary-extensions/-/binary-extensions-2.3.0.tgz",