You are a PMO manager , Do you coordinate with the HR department in developing career path system in your organization?

Yes, if necessary

Can intervene occasionally to specify the skills required for some professionals, as well as when they discover that some of the workers have the required level and need to be replaced

PMO manager may help in develop Job descriptions, roles and responsibilities for PMO staff, but not for whole the organization that is not  PMO manager work scope.

sure

Yes, major responsbilities I was responsible for as I was involved in setting in my last 3 companies - Bayer, DAD and PIC

Yes I do. One of the main roles of PMO is to identify the shortcomings in the skill sets of the related company's man power and prepare the personal and professional development plans for the staff and coordinate that with the HR department in preparing, implementing, and monitoring the performance of those plans.

No