Water is okay to use in a cars radiator but only as a temporary measure.
If you have a coolant leak and need to add water and have no coolant then using water in the radiator and engine to cool the engine down is better than running the engine without any water or coolant.
However long term use of water in the radiator can corrode radiators and aluminum engine blocks and even corrode water pumps.
If the engine is cast iron then the water can cause rust in the engine and fail to lubricate the water pump.
Coolant not only cools the engine it also lubricates the water pump and also prevents rust and corrosion.
Also coolant is antifreeze as well and if you live in a cold climate then your engine block can freeze and crack as well as your radiator and cause damage if you use just water.
So make sure you use coolant and antifreeze to prevent freezing of the engine block and radiator and also to prevent rust or corrosion.
Coolant or antifreeze can freeze if not mixed properly or if you live in an area where you get extremely cold temperatures such as below 30 F.
Yes it is possible for antifreeze to freeze up if the temperature gets cold enough.
Antifreeze will not freeze up at most normal cold weather temperatures as long as the antifreeze has been mixed properly in your engine.
If you put straight antifreeze in the engine then it will freeze at a temperature of between zero and minus five degrees Fahrenheit.
Adding water to the antifreeze makes the freezing point lower so it would take a colder temperature of below 30 F or so before the antifreeze froze up.
But as long as you mix the antifreeze properly or use the premixed antifreeze and coolant in your vehicle then the antifreeze or coolant should never freeze as long as you do not live in some place really, really cold.