There's no problem putting eggs straight into the fridge. The reason it's usually advised to let things cool is to put as little heat into the fridge as possible to prevent the internal temperature rising. It really is only rule of thumb advice to avoid putting in food from the oven which is at significantly higher temperature than the fridge.
When considering temperature everything is relative and the scale of temperature you should consider is degrees Kelvin not Celsius (Centigrade )or Fahrenheit which are scales offset to a norm which makes the numbers come closer to a range of numbers we can physically appreciate. So if you consider chicken bum temperature and room temperature in the absolute scale of degrees Kelvin, room temperature is about 295 degK and chicken bum temperature is 310 degK ie a mere 5% difference in absolute terms and the mass of an egg is pretty small so the amount of additional heat you add putting in the very warmest eggs is very little.
The in out of the fridge argument is a perennial lot of hot air. Cooks prefer to cook with room temperature ingredients mainly as it helps to give greater consistancy of cooking times rather than have ingredients at all sorts of different temperatures. The old myth about taste and/or nutrional value has been shown in many blind tasting trials to be just prejudice. Don't believe me you can do a double blind test yourself just ask me how.
The recommended temperature to keep eggs according to Food Standards agencies is to keep in a fridge. In fact in the USA all eggs are sold from the fridge which I believe is the law and in the UK official advice is to sell from the fridge but due to public preference there's mixed practise. On the continent it is usually room temperature. Eggs stay fresher much longer stored at fridge temperatures as with any food, this is a fact.
However eggs are "living" things which are often still viable to hatch even after 12 or more days at room temp. I have also hatched eggs (by accident) which have been stored in a fridge for a week, I've not tried it for longer periods. Much depends on the initial condition of the egg. Clean eggs with no dirt or cracks, from healthy hens will last at least two weeks at room temperature but as bacteria inside an egg will multiply rapidly at room temperature it's far safer to refridgerate.
I've seen it suggested even on this site that soft shelled eggs can be eaten which is a very risky practise indeed. If you've ever had food poisoning you wouldn't risk it believe me. If you are that desperate not to waste soft shelled eggs make sure they are thoroughly cooked before you feed them back to the chickens or put them in the bin. Or just make a souffle for the mother in law!
HF