chore: migrate to @vigdorov/* shared configs (eslint, prettier, typescript, vite)

Replace project-local ESLint, Prettier, TypeScript, and Vite configs
with shared packages from dev-configs monorepo.

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

backend/package.json

@@ -46,8 +46,6 @@
     "typeorm": "^0.3.28"
   },
   "devDependencies": {
-    "@eslint/eslintrc": "^3.2.0",
-    "@eslint/js": "^9.18.0",
     "@nestjs/cli": "^11.0.0",
     "@nestjs/schematics": "^11.0.0",
     "@nestjs/testing": "^11.0.1",
@@ -56,12 +54,10 @@
     "@types/node": "^22.10.7",
     "@types/passport-jwt": "^4.0.1",
     "@types/supertest": "^6.0.2",
+    "@vigdorov/eslint-config": "^1.0.1",
+    "@vigdorov/typescript-config": "^1.1.0",
     "eslint": "^9.18.0",
-    "eslint-config-prettier": "^10.0.1",
-    "eslint-plugin-prettier": "^5.2.2",
-    "globals": "^16.0.0",
     "jest": "^30.0.0",
-    "prettier": "^3.4.2",
     "source-map-support": "^0.5.21",
     "supertest": "^7.0.0",
     "ts-jest": "^29.2.5",