(1) Have the care and custody of all the county property, except such as is by law placed in the custody of other officers;

(2) Appoint an agent or attorney to take care of such property, and fix such agent's or attorney's compensation;

(3) Control all books, papers and instruments pertaining to the county mayor's office;

(4) Audit all claims for money against the county;

(5) Draw, without seal, all warrants upon the county treasury;

(6) Audit and settle the accounts of the county trustee, and those of any other collector or receiver of county revenue, taxes, or incomes, payable into the county treasury, and those of any persons entrusted to receive or expend any money of the county;

(7) Require such officers or persons to render and settle their accounts as directed by law, or by the authority under which they act;

(8) Enter in a book, to be known as the warrant book, in the order of issuance, the number, date, amount and name of the drawee of each warrant drawn upon the treasury;

(9) Keep in a suitable book an account of the receipts and expenditures of the county, so as to show clearly the assets of the county, and the debts payable to and by it, balancing the account semiannually, and generally to superintend the financial concerns of the county; and

(10) Report, in writing, semiannually, to the county legislative body all moneys received and paid out, and a complete statement of the financial condition of the county, and the county mayor shall settle the county mayor's accounts once every year.