If he's mating, then the answer is yes. Some cockerels start much younger than others, but they know better than we do whether they are old enough
Fertilized eggs will not grow chicks if you put them in the fridge. Refrigerating usually kills the embryo too, so they will not hatch if you later put them in an incubator or under a broody hen. (Chicks never develop inside the fridge, but on rare occasions someone has taken eggs out of the fridge, incubated them, and gotten chicks to develop and hatch.)
There is a range of temperature warmer than the fridge, up to about 75 degrees or so (Farenheit), where eggs just kind of pause: no chicks grow, but the embryos are still alive and can start growing if you put them somewhere warm enough (incubator, under broody hen).
If your "room temperature" is warmer than about 75-80 degrees, the embryos might start to grow. That's not warm enough to develop properly into healthy chicks, but they can start to grow and then die. (Which is not what you want to find in the eggs when you eat them!)