Sorry for the late response...
Addressing your main question: Generally speaking, I would follow my audit schedule. Therefore, I would start the audit process. Where surprise is not an issue, I would send advance notice to the auditee (two to four weeks ahead of time, depending upon the amount of request information needed, perceived scheduling issues, etc.). In your specific situation, your preliminary memo and request list will put the auditee on notice and could incent them to finish up their project. Depending upon the circumstances, I have occasionally agreed to a one time postponement of the audit if the auditee couldn't be ready, there were scheduling issues, etc. So you do have some flexibility.
First rabbit...I believe that the audit schedule should be a living document, not prepared once a year and set in stone. At any given time, ask yourself if what you are working on right now and in the coming weeks are the most important thing for you to be working on...if not, you may need to revise your audit schedule.
Second rabbit...coordination of audit effort. You are wise to see what the independent auditor says, not so much as to see if they found any deficiencies, but to avoid duplication of effort. It is a good idea to sit down with all external auditors/providers every year and go over the scope of their audits. Generally, internal audit does not need to cover procedures that will be addressed externally by an independent firm. In the real world, where audit resources are limited, we need to coordinate our effort where possible.
Third rabbit...tracking deficiencies. One best practice that I have seen over the years is to track all outstanding issues (from regulators, internal auditors, external auditors and consultants, etc.) in one central database. The database should indicate the area, source of recommendation, management response, manager responsible, original due date, revised due date (managers should be required to request an extension, with approval by exec management or the audit committee), status comments, priority, etc. By using a priority code, you can filter what gets reported to the audit committee (i.e. high level deficiencies), executive management (i.e. moderate and high level deficiencies), line management (all deficiencies and recommendations), etc. Managers should be required to update this database monthly, with monthly or quarterly reporting to management, and quarterly reporting to the audit committee. Using this approach, nothing should slip through the cracks.
_________________________
My opinions are just that...my opinions.