An egg requires temperatures over 90*F (ideally around 100*F but they will develop at slightly lower temperatures just more slowly) to develop. Now I admit, I have has some of my fertile eggs start to develop on my shelf - because they were in the sun during a hot week they easily hit over 90*. And sometimes we get air temps over 90*F during summer days, though my house usually stays a bit cooler.
But unless the egg is developing the shelflife remains about the same. You can fridge or not fridge. And eggs do keep just fine in a fridge, in fact they keep longer. I wash and fridge my eggs if I need them to keep for a long time (like once egg production slows down and I want eggs into the winter). Washed fresh eggs may keep in the fridge for many months. Otherwise I don't bother cause I like to cook with room temperature eggs for various reasons and I like to see them sitting out.
If the egg does develop, then stops, it tends to go bad very rapidly. So you should store fertile eggs under 90*F for results comparable to unfertilized eggs. But otherwise they're the same.