A general election for the elective officers of each city of the third class shall be held after the organization of the city under the provisions of this chapter and on municipal election days every two years thereafter, the city council may by ordinance provide for the nomination of officers by primary election; and provided, that all certificates of nomination and petitions therefor shall be filed with the city clerk and not with any other officer. Any city organizing under the provisions of this chapter may elect a mayor and may elect or provide for the appointment of such other officers as may be necessary to carry this chapter into effect, who shall hold office until their successors are elected or appointed and qualified; provided, that the assessor shall hold office until the first day of September and until his successor is elected or appointed and qualified, and the term of office of the assessor in such cities shall commence on the first day of September after the election at which he is elected or at such time as the city shall set the beginning of his appointive term, if appointed.