All appointments of officers and employees shall be without definite term, unless for temporary service not to exceed sixty days. Any officer or employee of the county appointed pursuant to § 15.2-512 may be suspended or removed from office or employment either by the board of county supervisors or the officer by whom he was appointed or employed. In case of the absence or disability of any officer, except the county clerk, the attorney for the Commonwealth, and the sheriff, which offices shall be filled as provided by general law, the board of county supervisors or other appointing power may designate some responsible person to perform the duties of the office.