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/service.py

@@ -76,6 +76,29 @@ async def get_messages(
     return await repository.list_chat_messages(db, chat_id, limit=safe_limit, before_id=before_id)
 
 
+async def search_messages(
+    db: AsyncSession,
+    *,
+    user_id: int,
+    query: str,
+    chat_id: int | None = None,
+    limit: int = 50,
+) -> list[Message]:
+    normalized = query.strip()
+    if len(normalized) < 2:
+        return []
+    safe_limit = max(1, min(limit, 100))
+    if chat_id is not None:
+        await ensure_chat_membership(db, chat_id=chat_id, user_id=user_id)
+    return await repository.search_messages(
+        db,
+        user_id=user_id,
+        query=normalized,
+        chat_id=chat_id,
+        limit=safe_limit,
+    )
+
+
 async def update_message(
     db: AsyncSession,
     *,