Yep, you're supposed to rest the bird for 2 to 3 days, to give rigor mortis time to pass. Check the bird, and when it's easier to bend the leg, you're done.
How cool you keep the bird has to do with sanitation. Too cold and rigor mortis will take longer (I think). So refrigerator temps are preferred for safety.
Others might know the details on how temperatures affect how long you have to rest the bird.
You didn't mention how old the rooster was. It's my understanding that young broilers don't need this resting.
The whole point is to avoid a too-tough bird after cooking. So if you were making stock, I don't think you'd need to rest it.
We culled a rooster a while back, and chose to just boil it down for soup and stock that same day. I don't think it would be necessary to rest a bird to do that.
You can also cook the bird before rigor mortis sets in, and avoid the resting process.
