Explanation
The most efficient method to insert the deleted row of a table into a history table for audit purposes is Trigger.
This is because a trigger is a database object that automatically executes a specified SQL statement when a certain event occurs on a table or view. A trigger can be defined to run before or after a delete operation on a table, and can access the deleted row using a special table called deleted. A trigger can perform the insertion of the deleted row into the history table without any additional logic or overhead from Appian. References: Triggers

Explanation

References:
* [Web APIs]
* [Connected Systems]
* [Expression Rules]
* [Decision Tables]

Explanation
Three methods that could be used to launch a test run for expression rule test cases when required are:
* A process model invoked via an API. A process model can be designed to run test cases for expression rules using the a!testRule() function or the Test Rule smart service. This process model can be exposed as a web API with an HTTP method such as POST or PUT, allowing external systems or applications to invoke it through an HTTP request.
* A process model exposed to users as an action. A process model can also be designed to run test cases for expression rules using the same function or smart service as above. This process model can be exposed to users as an action on an interface, such as a button or a link, allowing users to trigger it manually when needed.
* A SAIL interface embedded in a report. A SAIL interface can be created to run test cases for expression rules using the a!testRule() function. This interface can be embedded in a report, such as a grid or a chart, allowing users to view the test results interactively on an interface. References: Automated Testing for Expression Rules, a!testRule() Function, Test Rule Smart Service, Web APIs, SAIL Interfaces