Depends a lot on the context. How cold, how long, et cetera… but it’s possible.

Modern laying hens might lay an egg a day. When they’re clucky (wanting to raise some chicks) they will want to accumulate a week or so worth of eggs. During that time they’re not sitting on the nest, just saving up the eggs. Then when she’s ready she will start incubating them all at once.

This makes sense because if it takes n days for them to incubate she needs all the chicks to hatch within a day or else she won’t be able to feed the ones that hatched first while she’s still incubating the later ones.

As an aside, laying hens will just lay eggs with no rooster around, but quails generally don’t - they need a rooster around or they won’t produce. That being the case the majority of quail eggs you buy are fertilised while chicken eggs generally aren’t. IDK if it’s really true but there’s definitely claims that people have bought quail eggs at the supermarket and hatched quails at home.

Maybe. You’d have to have a fertilized egg, of course, and it is NOT the recommended way to do it, but it might remain viable despite initial refrigeration. You don’t want t kill the blastoderm. Here’s a paper:
https://www.sciencedirect.com/science/article/pii/S0032579119310119

Unlikely. The cold would have killed the developing chick (hence why an incubating bird can’t go away and come back later), so all it will do is go rotten.

Assuming the egg is fertilised. If it’s not, you might be looking at the second coming, if you somehow made it hatch.

Yes. Even with humans, some people keep their eggs refrigerated for use at a time to implant them when they’re ready for a baby.