Both statements are true within the PRINCE2 framework. Statement 1 is accurate as the Project Manager does have day-to-day control within the limits set by the Project Board for a management stage. Statement 2 is also correct because if a Team Manager forecasts a deviation beyond Work Package tolerances, it should indeed be referred to the Project Manager. References: PRINCE2 Foundation official study materials.

The definition of management in the context of PRINCE2 involves directing the execution of project tasks as agreed for the project. Management encompasses the actions and decisions necessary to ensure that the project's tasks are completed as planned, aligning with the agreed methods and standards.
PRINCE2 7 Foundation resources on project management principles.

The Quality Management Approach in PRINCE2 should provide guidance on how and when prototyping is to be used within the project. This approach outlines the quality techniques and standards to be applied and the responsibilities for achieving the required quality levels. Prototyping, as a method to validate product quality and user requirements early in the development cycle, falls under these guidelines.
(PRINCE2 7 Foundation Handbook, AXELOS)

If projects are not aligned with the corporate business strategy, a potential problem is that the objectives of different projects may overlap. This can lead to duplicated efforts, resource wastage, and conflicts between projects, all of which detract from organizational efficiency and effectiveness. References: PRINCE2 Foundation Handbook, Project Alignment with Corporate Strategy section.