If I collect the eggs each and every time they lay? Do they go broody without eggs visible in the nest box? Right now I'm collecting the eggs as soon as they jump down from the nest box.
Some people think a hen is more likely to go broody if the eggs build up in her nest I think there is something to this, but there are no guarantees. I've tried adding a golf ball a day to a nest and none went broody. I have had them go broody with no eggs at all in the nest. Whenever the hormones kick in, they go broody. What kicks off those hormones, who knows. There is little that is consistent with chickens.
Also, can I put the warm eggs in the fridge right away or should I wait until room tempature? Does it even matter?
It really does not matter. They should not raise the temperature of the other stuff in the refrigerator enough to cause a problem with that other stuff. As for the eggs themselves, it really does not matter.