For mobile app , after payment is instantiated , which method we should use to check success and failure response from 3rd party . And send order confirmation mail .
mobile order success action is which method for mobile app?
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If you are using soap, you can do it with http://www.magentocommerce.com/api/soap/sales/salesOrder/sales_order.info.html
But I wouldn't do this, because you don't want to give your users (inside of your app) the credentials to check all orders.
So the alternative is to implement your own api which logs in as the customer and then is only allowed to see their orders.