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/__init__.py

@@ -4,6 +4,8 @@ from app.services.game_manager import GameManager
 from app.services.room_manager import RoomManager
 from app.services.replay_manager import ReplayManager
 from app.services.achievement_manager import AchievementManager
+from app.services.timer_manager import TimerManager, timer_manager
+from app.services.question_service import QuestionService, question_service
 
 __all__ = [
     "AIValidator",
@@ -12,4 +14,8 @@ __all__ = [
     "RoomManager",
     "ReplayManager",
     "AchievementManager",
+    "TimerManager",
+    "timer_manager",
+    "QuestionService",
+    "question_service",
 ]