Cool air enters the radiator and becomes hot. Won't the hot air now flow over the engine and make the engine hotter too?
The difference is that a radiator is constructed to have very high surface area it is this contact area between the surface of the radiator and the air which allows a high rate of heat transfer.
The other element is that in air entering the radiator is cold, heat transfer rate is proportional to temperature difference so if the coolant flowing through the radiator is at 90 deg C then the air might be heated to maybe 50C but certainly not more than 90 so it is still cooler than the engine.
No it won't. Air needs somewhere to go, and the engine compartment doesn't have much of an air outlet. Sure, a teeny-tiny little bit of hot air will go over the engine, but the engine is plenty hot on it's own already.
The radiator is hitting a wall of air in front of it, it's not really the airflow through it that is providing the cooling as such.