It's definitely more likely than a mixed-sex chicken, but I'm not sure why a rat would carry an egg all the way to a different coop only to not eat it.
If you hear galloping outside, do you think "zebra" or "horse"? I'm not coming to conclusions yet, but there're lots of possibilities which are more probable than a "hooster" (hen-rooster).
Those two eggs are completely different colors. Sounds like somewhere in this chain of events there's a prankster. You sure nobody's been putting eggs in there?