1) If the leasing period is full, and the lease had no section to convert the leasing term to month by month, then your next month payment automatically renews the lease to original term if you did not give written notice to terminate the lease;
2) The late fee charg is solid, you have no case. However, it sounds the charge rate is too high, you may talk with a real estate attorny in your state to see any limit to the late fee charg;
This sounds not right too, check with your attorny: Lease Section 8 "Application of Funds" states: "Landlord will apply all funds received from Tenant first to any non-rent obligations of Tenant including late charges, returned check charges, charge-backs for repairs, brokerage fees, and periodic utilities, then to rent."