According to the PMBOK Guide, 7th edition, one of the best practices for introducing agile approaches in an organization is to start with a pilot project that has a clear scope, a committed sponsor, and a willing team.
This way, the project manager can demonstrate the benefits of agile, gain feedback, and learn from the experience. The project manager should also provide agile training to the team members and stakeholders who are involved in the pilot project, to ensure they understand the agile principles, values, and practices. References: PMBOK Guide, 7th edition, page 23; Agile Practices and the PMP Certification

To enhance the budget estimations for a software project, especially for a major recurring customer, it is essential to review budget-related lessons learned from similar projects. This approach allows the project manager to utilize historical data and experiences to inform the current project's budget estimation. By analyzing past projects, the project manager can identify trends, anticipate potential issues, and adjust the budget accordingly to avoid previous pitfalls. This method aligns with the best practices of project cost management as outlined in the Project Management Body of Knowledge (PMBOK Guide), which emphasizes the importance of historical information and lessons learned for accurate budget planning.
: (Project Management Institute's PMBOK Guide and Professional in Business Analysis Reference Materials)

A customer who wants to show a tangible outcome faster is likely to have a high value for early and frequent delivery, which is a principle of agile and adaptive approaches. In such a situation, the project manager should review the customer's priorities regarding the desired requirements for the milestones, and discuss which requirements can be delivered faster with the project team. This is a way to apply the concept of value-driven delivery, which focuses on maximizing the value of the product and satisfying the customer's needs.
Informing the customer that moving the milestone has a very high impact on project cost, escalating the issue to senior management, or convincing the customer to keep the requirements as is are not effective ways to deal with the customer's request, as they do not address the customer's value or needs, and may create conflict or dissatisfaction. References: [PMI Agile Practice Guide], page 8; [PMBOKGuide 6th Edition], page 18.

In an iterative project, when a new compliance requirement is introduced that impacts the project, it is essential to update the project documentation to reflect this change. The product backlog is a living document that captures all the requirements, including new and changed ones, that need to be addressed in the project.
By updating the product backlog, the project manager ensures that the team is aware of the new compliance requirement and can plan for its implementation in upcoming iterations. This action aligns with the agile principle of welcoming changing requirements, even late in development, as a means to enhance customer competitive advantage (as per the Agile Manifesto).
: = The answer is based on the best practices outlined in the PMBOK Guide and the Agile Practice Guide, which are part of the PMP certification materials. These guides emphasize the importance of maintaining an up-to-date product backlog and incorporating changes that may arise during the project lifecycle.

Team building in project management is the process of developing a cohesive team that will work together to achieve successful outcomes12. It involves identifying the strengths and weaknesses of each member and using those insights to create an environment where collaboration, trust, and open communication are encouraged3. In this case, facilitating a team-building meeting can help address the issue of team members working individually rather than as a team. This meeting can help the team members understand each other better, build trust, and foster a sense of belonging, which can encourage them to work together as a team12.
References:
Team Building in Project Management | PMI1
Team Building | Project Management Professional (PMP) - GreyCampus2
Ways to Create a Strong Project Team | PMI - Project Management Institute4 Team Building: Success Path to Project Management - Eduhubspot3