Initial extraction: groombook/api from groombook/app monorepo

Part of GRO-802 monorepo breakdown.

Changes:
- Extract apps/api/ as the main API service
- Inline packages/db/ (database schema, migrations, utilities)
- Inline packages/types/ (shared TypeScript types)
- Add CI workflow for lint, typecheck, test, build, docker
- Port Dockerfile with 4 stages: runner, migrate, seed, reset

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

apps/api/eslint.config.js

@@ -0,0 +1,11 @@
+import tseslint from "typescript-eslint";
+
+export default tseslint.config(
+  ...tseslint.configs.recommended,
+  {
+    rules: {
+      "@typescript-eslint/no-explicit-any": "warn",
+      "@typescript-eslint/no-unused-vars": ["error", { argsIgnorePattern: "^_" }],
+    },
+  }
+);