The rooster (닭, dak) is read almost entirely through timing — its traditional role announcing the dawn makes it a symbol of news arriving right on schedule, or a wake-up call arriving exactly when it's needed, rather than a symbol tied to a fixed fortune the way the pig or ox are.
A rooster crowing clearly and confidently is read as good news arriving punctually — not a surprise windfall, but something you may have already been expecting finally showing up. This pairs naturally with the rooster's real-world role: nobody's shocked when the rooster crows at dawn, and the dream inherits that same sense of an announced, anticipated arrival.
A hen laying eggs is read as new opportunities in the process of "hatching" — plans or wealth that are already set in motion but haven't fully arrived yet, a patient, building version of fortune rather than an instant one. Catching a rooster before it can get away is read as seizing an opportunity right as it's about to be announced publicly, before anyone else gets to it.
A rooster crowing at the wrong time — outside its usual dawn moment — is read more cautiously by some interpreters, as news arriving at an inconvenient or unexpected moment, worth being ready for even if it isn't clearly good or bad on its own.