Flush your cooling system with Citric acid, change all ribber pipes and hoses and drive on water for a couple of days. If everything remains ok, drain and fill with coolant.
The stock 4K engine in 1982 Corolla comes with a belt driven fan.
Use Shell HX7 or HX5 for the engine. If the fuel pump is ok, leave it as it is.
For manual transmission, check which gearbox is installed and then Google the recommended grade.