Recently, when it is hot outside, my temperature warning light comes on. Now I have antifreeze with water mix in bowl, I'm not loosing it. My fan is spinning when A/C is on.

If the light is coming on when the day is hot then the probability is that there is something wrong with your cooling system. If it came on randomly then I would suspect a temp sensor. Usual suspects are the water pump, radiator and the thermostat. The thermostat can be inspected easily. The water pump requires removal of the timing belt and the radiator needs to be volume/flow tested. The thermostat can stick closed, the fins on the water pump can disintegrate and the radiator can gunk up with calcium.
Hmm not sure what you mean there. Do you mean the fan doesn't run on when you turn the ignition off? It's not supposed to.
Isnt fans job when motor gets hot to cool it,along with cooliant?
Hmm not sure what you mean there. Do you mean the fan doesn't run on when you turn the ignition off? It's not supposed to.
Yes, but the fan will not run with the ignition OFF. Is that what you mean!? I don't understand what you mean by "turn of engine". Do you mean turn off engine, or what?