Mostly bad eggs SMELL. and the yolk kinda goes brown, if I remember correctly... it's been a while... but you'll know by the sulfur/rotten egg smell.
They should be okay for a few weeks. The fridge v. the counter debate, as far as I can tell, still isn't settled. I was working on a boat for 5 weeks (no shore/supply stops) and the cook swore that they kept better in cool, dry storage than a refrigerator. She kept our eggs in the ship pantry. We had crates and crates of eggs, and I think we had our last egg dish in week 4 of the trip. By that time, we were cracking them into a separate bowl, tossing the rotten ones and then transferring the good ones to another bowl... I still remember tossing rotten eggs overboard in the middle of the ocean, what a surreal experience!
In other countries, eggs aren't usually sold in the refrigerator section, but on shelves like cereal. Still, some people say that refrigerating them helps them keep longer. So -- it depends on whether you have more space in your fridge or on your counter! I know my counters are always packed...