fix: resolve all typecheck, lint, and test failures

- Add @types/node to packages/db devDependencies (typecheck was missing process)
- Re-export drizzle-orm helpers (eq, gte, etc.) from @groombook/db to avoid
  duplicate-instance type conflicts; remove drizzle-orm direct dep from API
- Add @hono/zod-validator and jose as direct API dependencies
- Merge duplicate @groombook/db imports in all route files
- Fix noUncheckedIndexedAccess errors: appointments PATCH, web calendar grid
- Fix weightKg/dateOfBirth type conversion in pets route (numeric→string, string→Date)
- Add eslint.config.js for API and web (ESLint 9 flat config format)
- Add vitest.config.ts with passWithNoTests for API and web

Co-Authored-By: Paperclip <noreply@paperclip.ing>

apps/api/package.json

@@ -15,7 +15,9 @@
     "@groombook/db": "workspace:*",
     "@groombook/types": "workspace:*",
     "@hono/node-server": "^1.13.7",
+    "@hono/zod-validator": "^0.4.3",
     "hono": "^4.6.17",
+    "jose": "^5.9.6",
     "openid-client": "^6.1.7",
     "zod": "^3.24.1"
   },