What is the authenticity of the mentioned hadith, and is supplicating upon hearing the crow of a rooster and the braying of a donkey restricted to nighttime only, and what is the specific supplication when hearing the crow of a rooster?

The hadith is authentic and has been reported in an absolute sense, not restricted to the night. Most scholars have understood its generality and have not limited it to the night. They have recommended making supplication upon hearing the crowing of roosters, by generally asking Allah for His bounty, and not by a specific supplication. It is reported in one narration: "Then ask Allah and yearn towards Him." Some scholars are of the view that hearing the crowing of roosters during the night is a time when supplications are answered.

It does not mean that roosters crow only when they see an angel or a devil, but their crowing might also be for that reason. Therefore, it is recommended to supplicate and seek refuge [with Allah] with every crowing.