feat(api): add global error handler with typed error classes

- Add AppError base class, ValidationError (400), NotFoundError (404)
- Add central error middleware in app.ts
- Remove inline safeParse error handling from controllers
- Replace plain Error throws with NotFoundError in gameService

apps/api/src/middleware/errorHandler.ts

@@ -0,0 +1,18 @@
+import type { Request, Response, NextFunction } from "express";
+import { AppError } from "../errors/AppError.js";
+
+export const errorHandler = (
+  err: Error,
+  _req: Request,
+  res: Response,
+  _next: NextFunction,
+) => {
+  if (err instanceof AppError) {
+    res.status(err.statusCode).json({ success: false, error: err.message });
+    return;
+  }
+
+  console.error("Unexpected error:", err);
+
+  res.status(500).json({ success: false, error: "Internal server error" });
+};