feat: add search APIs and telegram-like chats sidebar flow

- implement chat query filtering and message search endpoints

- add db indexes for search fields

- activate chats search input in web

- replace inline create panel with floating TG-style action menu

app/chats/router.py

@@ -31,10 +31,11 @@ router = APIRouter(prefix="/chats", tags=["chats"])
 async def list_chats(
     limit: int = 50,
     before_id: int | None = None,
+    query: str | None = None,
     db: AsyncSession = Depends(get_db),
     current_user: User = Depends(get_current_user),
 ) -> list[ChatRead]:
-    return await get_chats_for_user(db, user_id=current_user.id, limit=limit, before_id=before_id)
+    return await get_chats_for_user(db, user_id=current_user.id, limit=limit, before_id=before_id, query=query)
 
 
 @router.post("", response_model=ChatRead)