Putting food from the oven to the refrigerator is perfectly safe and often recommended. The only caveat would be not to place food that is in a very hot pan directly into the refridgerator. Transfer the food to a different container.
You should allow the food to cool to atleast room temperature before you put it in your refridgerator, if you put it in the firdge hot, it will lower the internal temperature of your refridgerator.
This is a controversial one but I say yes. It is better to warm up the refirgeratoe a little than risk food borne illnesses with food left on the counter to cool.
