refactor(phase3-session-a): extract Image Studio models and deps into separate modules

- Created routers/image_studio/models.py with all 40 Pydantic models
- Created routers/image_studio/deps.py with get_studio_manager() and _require_user_id()
- Renamed old monolithic image_studio.py -> image_studio_router.py
- Updated __init__.py to re-export the router for backward compatibility
- Old file now imports models and deps from new modules (no inline definitions)

Backward compatibility: from routers.image_studio import router still works.
Route count unchanged: 33 routes, prefix /api/image-studio.

backend/routers/image_studio/deps.py

@@ -0,0 +1,35 @@
+"""Shared dependencies for Image Studio API endpoints."""
+
+from typing import Dict, Any
+from fastapi import Depends, HTTPException, status
+
+from services.image_studio import ImageStudioManager
+from middleware.auth_middleware import get_current_user
+from utils.logger_utils import get_service_logger
+
+logger = get_service_logger("api.image_studio")
+
+
+def get_studio_manager() -> ImageStudioManager:
+    """Get Image Studio Manager instance."""
+    return ImageStudioManager()
+
+
+def _require_user_id(current_user: Dict[str, Any], operation: str) -> str:
+    """Ensure user_id is available for protected operations."""
+    user_id = (
+        current_user.get("sub")
+        or current_user.get("user_id")
+        or current_user.get("id")
+        or current_user.get("clerk_user_id")
+    )
+    if not user_id:
+        logger.error(
+            "[Image Studio] ❌ Missing user_id for %s operation - blocking request",
+            operation,
+        )
+        raise HTTPException(
+            status_code=status.HTTP_401_UNAUTHORIZED,
+            detail="Authenticated user required for image operations.",
+        )
+    return user_id