The purpose of the Initiating a Project process is to establish solid foundations for the project, ensuring that there is a clear understanding of the reasons for the project, the benefits expected, and how the delivery will be managed. This foundation is crucial for the successful management and delivery of the project.
References: PRINCE2 Foundation manual, PRINCE2 official resources

The purpose of the 'controlling a stage' process in PRINCE2 is to ensure that the stage stays on course and reacts to unexpected events. This includes taking corrective actions when there are deviations from the stage plan that can be managed within the tolerances set by the project board. References: PRINCE2 Foundation Handbook, Controlling a Stage section.

Risk response refers to the actions identified and implemented to bring the risk exposure to an acceptable level. These actions are part of the risk management process and aim to reduce, avoid, transfer, or accept risks depending on their potential impact on the project. References: PRINCE2 Foundation Training Manual; PRINCE2 2017 Foundation handbook.

This answer remains the same as question 3. The Benefits Review Plan's purpose does not typically include defining the period for the cost-benefit analysis. Instead, it focuses on outlining how the benefits of the project are to be reviewed and measured after the project's closure. References: PRINCE2 Foundation official guide and study material.
These answers are derived from the official PRINCE2 documentation and study materials available for the PRINCE2 Foundation certification.

The process that links the Team Manager role with the PRINCE2 method in a commercial environment is
'Managing Product Delivery'. This process ensures that the project's products are created and delivered efficiently and effectively, aligning the work of the team with the overall project management approach.
References: PRINCE2 7 Foundation Handbook and official training materials.