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

scripts/gametest/test-game.ts

@@ -34,6 +34,25 @@ async function main() {
       `${question.prompt}: ${result.data.isCorrect ? "✓" : "✗"} (picked ${0}, correct was ${result.data.correctOptionId})`,
     );
   }
+
+  const badRequest = await fetch("http://localhost:3000/api/v1/game/start", {
+    method: "POST",
+    headers: { "Content-Type": "application/json" },
+    body: JSON.stringify({ source_language: "en" }),
+  });
+  console.log("400 test:", badRequest.status, await badRequest.json());
+
+  // Send a valid shape but a session that doesn't exist
+  const notFound = await fetch("http://localhost:3000/api/v1/game/answer", {
+    method: "POST",
+    headers: { "Content-Type": "application/json" },
+    body: JSON.stringify({
+      sessionId: "00000000-0000-0000-0000-000000000000",
+      questionId: "00000000-0000-0000-0000-000000000000",
+      selectedOptionId: 0,
+    }),
+  });
+  console.log("404 test:", notFound.status, await notFound.json());
 }
 
 main();