What is the ruling on raising one's voice in sending blessings upon the Prophet (peace be upon him) and in expressing satisfaction with the Rightly Guided Caliphs between the rak'ahs of Tarawih prayers?

There is no basis for this in Islamic law; rather, it is one of the innovated heresies. It is obligatory to abandon it. The rectification of the nation is achieved by following the Qur’an and the Sunnah, and what the predecessors of the Ummah followed, and by being wary of anything that contradicts that.