refactor(gateway): restructure gateway with new dependency injection and services

Reorganize app/gateway/ with:
- common/ - lifespan management
- dependencies/ - FastAPI dependency injection (db, checkpointer, repositories, stream_bridge)
- services/runs/ - run execution services (facade_factory, input adapters, store operations)
- registrar.py - router registration
- router.py - main router setup

Simplify app.py to use the new modular structure.
Remove deprecated utils.py.

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

backend/app/gateway/common/__init__.py

@@ -0,0 +1,3 @@
+from .lifespan import lifespan_manager
+
+__all__ = ["lifespan_manager"]

backend/app/gateway/common/lifespan.py

@@ -0,0 +1,52 @@
+from collections.abc import Callable
+from contextlib import AbstractAsyncContextManager, AsyncExitStack, asynccontextmanager
+from typing import Any
+
+from fastapi import FastAPI
+
+LifespanFunc = Callable[[FastAPI], AbstractAsyncContextManager[dict[str, Any] | None]]
+
+
+class LifespanManager:
+    """FastAPI lifespan manager"""
+
+    def __init__(self) -> None:
+        self._lifespans: list[LifespanFunc] = []
+
+    def register(self, func: LifespanFunc) -> LifespanFunc:
+        """
+        Register a lifespan hook.
+
+        :param func: lifespan hook
+        :return:
+        """
+        if func not in self._lifespans:
+            self._lifespans.append(func)
+        return func
+
+    def build(self) -> LifespanFunc:
+        """
+        Build the combined lifespan hook.
+
+        :return:
+        """
+
+        @asynccontextmanager
+        async def combined_lifespan(app: FastAPI):  # noqa: ANN202
+            state: dict[str, Any] = {}
+            async with AsyncExitStack() as exit_stack:
+                for lifespan_fn in self._lifespans:
+                    result = await exit_stack.enter_async_context(lifespan_fn(app))
+                    if isinstance(result, dict):
+                        state.update(result)
+
+                for key, value in state.items():
+                    setattr(app.state, key, value)
+
+                yield state or None
+
+        return combined_lifespan
+
+
+# Singleton lifespan_manager instance
+lifespan_manager = LifespanManager()