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/errors/AppError.ts

@@ -0,0 +1,21 @@
+export class AppError extends Error {
+  public readonly statusCode: number;
+
+  constructor(message: string, statusCode: number) {
+    super(message);
+    this.name = this.constructor.name;
+    this.statusCode = statusCode;
+  }
+}
+
+export class ValidationError extends AppError {
+  constructor(message: string) {
+    super(message, 400);
+  }
+}
+
+export class NotFoundError extends AppError {
+  constructor(message: string) {
+    super(message, 404);
+  }
+}