Does Zerodha Send SMS Alerts or Notifications for Trades?

No, Zerodha doesn't send SMS notifications for trades, but exchanges do. You need to follow the below steps to get trade alerts on Kite: 

Steps to get trade alerts on Kite: 

Steps	Description
1	Firstly, tap on Client ID.
2	Then, tap on Settings.
3	Finally, tap on Order Notifications.

How Do I Get Trade Alerts on Kite?

Step 1: Firstly, tap on Client ID.

Step 2: Next, tap on Settings.

Step 3: Finally, tap on Order Notifications.

You don't need to keep the Kite App running in the background to receive these notifications.