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/vitest.config.ts

@@ -0,0 +1,7 @@
+import { defineConfig } from "vitest/config";
+
+export default defineConfig({
+  test: {
+    passWithNoTests: true,
+  },
+});