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/messages/router.py

@@ -4,7 +4,7 @@ from sqlalchemy.ext.asyncio import AsyncSession
 from app.auth.service import get_current_user
 from app.database.session import get_db
 from app.messages.schemas import MessageCreateRequest, MessageRead, MessageStatusUpdateRequest, MessageUpdateRequest
-from app.messages.service import create_chat_message, delete_message, get_messages, update_message
+from app.messages.service import create_chat_message, delete_message, get_messages, search_messages, update_message
 from app.realtime.schemas import MessageStatusPayload
 from app.realtime.service import realtime_gateway
 from app.users.models import User
@@ -27,6 +27,17 @@ async def create_message(
     return message
 
 
+@router.get("/search", response_model=list[MessageRead])
+async def search_messages_endpoint(
+    query: str,
+    chat_id: int | None = None,
+    limit: int = 50,
+    db: AsyncSession = Depends(get_db),
+    current_user: User = Depends(get_current_user),
+) -> list[MessageRead]:
+    return await search_messages(db, user_id=current_user.id, query=query, chat_id=chat_id, limit=limit)
+
+
 @router.get("/{chat_id}", response_model=list[MessageRead])
 async def list_messages(
     chat_id: int,