Has it been narrated from the Prophet, peace and blessings be upon him, a ḥadīth: "If the sky is pouring and the earth is gushing"? And are there any ḥadīth that confirm or deny combining prayers due to rain?

It has not been reported from the Prophet, peace be upon him, anything indicating that the saying, "If the sky is 'munbalah' (pouring heavily)" is a hadith. However, there are many hadiths that permit combining prayers due to rain, among them the hadith of Ibn Abbas in Bukhari, that the Prophet, peace be upon him, prayed in Medina seven and eight rak'ahs: Dhuhr and Asr, and Maghrib and Isha. Ibn Umar also combined prayers with the rulers during rain. Similarly, Urwah ibn al-Zubayr, Sa'id ibn al-Musayyab, Abu Bakr ibn Abd al-Rahman, and Umar ibn Abd al-Aziz would combine Maghrib and Isha prayers on a rainy night.