It is for the Council, not the parent Committee, to determine whether the public may attend a Sub-Committee meeting and any public notice requirements. In order to facilitate proper decision-making, I suggest that members receive at least three days' notice with full documentation.
Having said this, I'd be tempted to dispense with the Sub-Committee. The terms of reference of the Staffing Committee should delegate full responsibility for routine staffing matters to the Committee and any grievance or disciplinary issues should be handled in accordance with those policies. The annual appraisal of the Clerk is best carried out by two suitably qualified Councillors, who can be appointed by your Staffing Committee. There is a tendency to specify Chair and Vice-Chair or Chair and one other, but if these named individuals are not the most capable, they should not undertake such an important role. The appraisers report to the Staffing Committee and may make recommendations as appropriate.
Committee meetings should be run in the same way as full Council meetings, with the Clerk in attendance, but if the agenda item relates to the Clerk, they should be asked to step out of the meeting, in the same way that a Councillor with a pecuniary interest would. One of the Councillors should then take notes to comprise the minutes for that item and it is helpful for the exact wording of the minutes to be agreed before the Clerk is readmitted. Minutes should be nothing more than a note of decisions taken, so the detail of the discussion remains confidential to the participants.