Is it established that the Prophet (peace be upon him) did not raise his hands at certain times during bowing (rukuʿ) and rising from it when saying: "Allah is the Greatest" and "Allah hears those who praise Him"? And is there a reason or story behind raising the hands?

The raising of the hands is authentically established from the Prophet, peace and blessings be upon him, in four positions during prayer: when pronouncing the opening takbir (takbirat al-ihram), when bowing (ruku'), when rising from bowing, and when standing up from the first tashahhud. This is based on the hadith of Ibn Umar in Sahih al-Bukhari. Hands are not raised when prostrating (sujud) or when rising from prostration.

The raising of hands in prayer has multiple wisdoms, among them: exalting Allah Almighty, following the Messenger of Allah, peace and blessings be upon him, showing humility and submission, signaling the renunciation of worldly matters, engaging in prayer and conversing with Allah, and indicating entry into prayer.