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/app.ts

@@ -1,11 +1,14 @@
 import express from "express";
 import type { Express } from "express";
 import { apiRouter } from "./routes/apiRouter.js";
+import { errorHandler } from "./middleware/errorHandler.js";
 
 export function createApp() {
   const app: Express = express();
+
   app.use(express.json());
   app.use("/api/v1", apiRouter);
+  app.use(errorHandler);
 
   return app;
 }