According to the hybrid project management approach, which combines elements of waterfall and agile methodologies, the project manager should facilitate the collaboration and communication among the team members and adjust the scope and schedule as needed12. In this scenario, the project manager should first determine whether other cross-functional team members are able to take over the work of the developer who left the company in the middle of an iteration. This would allow the project to continue with minimal disruption and maintain the quality and value of the deliverable. If no one on the team has the skills or capacity to complete the work, the project manager should then consult with the product owner and the stakeholders to renegotiate the scope and schedule of the iteration, or to cancel the iteration and return to planning3. The project manager should avoid reviewing the backlog to reschedule the deliverable for a later iteration without involving the team and the product owner, as this would violate the agile principle of customer collaboration over contract negotiation4. The project manager should also avoid engaging with stakeholders directly to review the iteration process without involving the team and the product owner, as this would violate the agile principle of self-organizing teams4. The project manager should also avoid going to management and asking to hire a new external developer without exploring other options first, as this would incur additional costs and risks, and delay the delivery of the product feature5. References: 1: What is the Hybrid Approach in Project Management? | Hygger.io 2: Hybrid Project Management: What is it, Approach
& Software - Teamhood 3: What to do when capacity is reduced in the middle of the Sprint? (e.g. Sick Leave)
- Project Management Stack Exchange 4: Manifesto for Agile Software Development 5: How To Handle A Key Team Member Leaving At A Critical Project Point | IT Staffing & Technical Consulting

The project manager should first determine if the feature is within the scope of the project. This is because adding a new feature could potentially impact the project's budget, schedule, and other resources. If the feature is within the scope, it can be considered for inclusion. However, if it is outside the scope, a change request would need to be submitted and approved before the feature can be added12.
References:
4 Phases of the Project Management Lifecycle Explained
5 Phases of Project Management Life Cycle | Complete Guide.

When a stakeholder requests changes that could impact the project scope, the project manager should first review and validate the scope management plan. This plan outlines how scope changes are managed and controlled. If the new ideas are not covered within the existing project scope, they may require a formal change request and impact analysis before proceeding. This ensures that any changes align with project objectives and do not adversely affect project constraints such as time, cost, and quality.
References:
Project Management Institute. (2017). A Guide to the Project Management Body of Knowledge (PMBOK Guide) - Sixth Edition.
Project Scope Management according to the PMBOK1.
The PMBOK's Project Scope Management Knowledge Area2.
Project scope management: Plan, template, and guide3.

When a director demands the ability to make the final go/no-go decision about whether the systems go live, the project manager should schedule a meeting with the project sponsor to determine if it is appropriate to accept the director's request. The project sponsor is typically a senior executive in a corporation (often at or just below board level) who is responsible to the business for the success of the project. This meeting would help clarify roles and responsibilities, and ensure that the decision-making process aligns with the project's governance structure. References:
https://www.coursera.org/articles/project-management-lifecycle
https://www.projectmanager.com/blog/project-director-role