feat(gateway): dual-mount legacy routers on /api and /api/v1 (Stage 1 PR5)

Strip /api prefix from 13 legacy router APIRouter() declarations and dual-mount
each on prefix="/api" (backward compat) and prefix="/api/v1" (versioned surface)
in app.py. Auth, service-accounts, api-keys, assistants-compat remain single-mount.
Update 10 test files to pass prefix="/api" when directly including stripped routers.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

backend/tests/test_threads_router.py

@@ -59,7 +59,7 @@ def _build_thread_app() -> tuple[FastAPI, InMemoryStore, InMemorySaver]:
     app.state.store = store
     app.state.checkpointer = checkpointer
     app.state.thread_store = _PermissiveThreadMetaStore(store)
-    app.include_router(threads.router)
+    app.include_router(threads.router, prefix="/api")
     return app, store, checkpointer
 
 
@@ -113,7 +113,7 @@ def test_delete_thread_route_cleans_thread_directory(tmp_path):
     (paths.sandbox_work_dir("thread-route", user_id=user_id) / "notes.txt").write_text("hello", encoding="utf-8")
 
     app = make_authed_test_app()
-    app.include_router(threads.router)
+    app.include_router(threads.router, prefix="/api")
 
     with patch("app.gateway.routers.threads.get_paths", return_value=paths):
         with TestClient(app) as client:
@@ -128,7 +128,7 @@ def test_delete_thread_route_rejects_invalid_thread_id(tmp_path):
     paths = Paths(tmp_path)
 
     app = make_authed_test_app()
-    app.include_router(threads.router)
+    app.include_router(threads.router, prefix="/api")
 
     with patch("app.gateway.routers.threads.get_paths", return_value=paths):
         with TestClient(app) as client:
@@ -141,7 +141,7 @@ def test_delete_thread_route_returns_422_for_route_safe_invalid_id(tmp_path):
     paths = Paths(tmp_path)
 
     app = make_authed_test_app()
-    app.include_router(threads.router)
+    app.include_router(threads.router, prefix="/api")
 
     with patch("app.gateway.routers.threads.get_paths", return_value=paths):
         with TestClient(app) as client: