The register of deeds shall have the custody of and safely keep and preserve all books, records, maps, and papers kept or deposited in his or her office. He or she shall also record or cause to be recorded all deeds, mortgages, instruments, and writings presented to him or her for recording and left with him or her for that purpose. Plats and subdivisions are not authorized to be recorded if such plat or subdivision has not been approved by the city council, the village board of trustees, the agent of a city of the first or second class or of a village designated pursuant to section 19-916, or the governing body of the county, whichever is appropriate. When such deeds, mortgages, instruments, and writings are so recorded, it shall be the duty of the register of deeds to proofread or cause to be proofread such records. If an error should occur in recording any of the writings mentioned in this section thereby necessitating the rerecording of same, the expense thus incurred shall be paid out of the general fund of the county in the same way as any other claim, and the amount so paid shall be collected from the official responsible for the error or from his or her official bond. The register of deeds shall prepare and file the required annual inventory statement of county personal property in his or her custody or possession as provided in sections 23-346 to 23-350.