Is it obligatory for the worshipper to hear himself say the opening takbir (takbirat al-ihram)? And if he did not hear himself, is he required to repeat the past prayers?

The opening Takbir (Takbirat al-Ihram) is a pillar without which the prayer is not valid. The jurists have differed regarding the minimum sufficient articulation of it. The Shafi'is and Hanbalis are of the view that the minimum sufficient articulation is for the worshiper to hear himself. The Malikis are of the view that it is recommended to say it aloud to the extent that the worshiper hears himself and those next to him. If he does not do so and says it silently, he performs the prostration of forgetfulness (sujud al-sahw) as a recommendation, and his prayer is valid. Some jurists are of the view that it is sufficient to move the tongue with the letters, even if he does not hear them. This is the choice of Shaykh al-Islam Ibn Taymiyyah and one opinion among the Malikis. Accordingly, if one did not hear himself during the opening Takbir, his prayer is valid and he is not required to repeat it.