feat(phase3): Implement complete game logic with WebSocket events

Timer System:
- Add TimerManager service with asyncio for server-side timers
- Support steal time reduction (50% time)
- Automatic timer cancellation on answer

Question Loading:
- Add QuestionService to load daily questions from PostgreSQL
- Generate 8×5 board (categories × difficulties)
- Filter by date_active and approved status

Database Integration:
- Create GameSession in PostgreSQL when game starts
- Update scores during game and on finish
- Store db_session_id in Redis for cross-reference

Replay Integration:
- Save all game events: question_selected, answer_submitted, steal_attempted, steal_passed, game_finished
- Generate unique replay code on game finish

Achievement Integration:
- Initialize PlayerStats in Redis when joining room
- Update stats on every answer (streak, category, speed, etc.)
- Check achievements on game finish for all players

Game Finish:
- Automatic finish when all questions answered
- Manual finish by host
- Emit game_finished with winner, scores, replay_code, achievements

Phase 3 tasks completed:
- F3.1: Timer manager with asyncio
- F3.2: Question service for board loading
- F3.3: GameSession PostgreSQL integration
- F3.4: Replay event saving
- F3.5: Achievement stats tracking
- F3.6: Complete game finish flow

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

backend/app/services/room_manager.py

@@ -168,6 +168,38 @@ class RoomManager:
             return json.loads(data)
         return None
 
+    async def get_player_stats(self, room_code: str, player_name: str) -> Optional[dict]:
+        """Obtiene stats de un jugador."""
+        await self.connect()
+        data = await self.redis.get(f"stats:{room_code}:{player_name}")
+        if data:
+            return json.loads(data)
+        return None
+
+    async def set_player_stats(self, room_code: str, player_name: str, stats: dict) -> None:
+        """Guarda stats de un jugador."""
+        await self.connect()
+        await self.redis.setex(
+            f"stats:{room_code}:{player_name}",
+            3600 * 3,
+            json.dumps(stats)
+        )
+
+    async def init_player_stats(self, room_code: str, player_name: str) -> dict:
+        """Inicializa stats para un nuevo jugador."""
+        stats = {
+            "player_name": player_name,
+            "current_streak": 0,
+            "total_correct": 0,
+            "total_steals": 0,
+            "successful_steals": 0,
+            "category_correct": {},
+            "fastest_answer_seconds": None,
+            "questions_500_correct": 0
+        }
+        await self.set_player_stats(room_code, player_name, stats)
+        return stats
+
 
 # Singleton instance
 room_manager = RoomManager()