Does raising one's voice with weeping and supplication, and using unproven phrases during prostration—due to extreme distress—invalidate the prayer? And does the ruling differ between obligatory and supererogatory prayers?

The prayer is not invalidated by raising one's voice in supplication during prostration, nor is it invalidated by weeping for one who is overcome by it, whether out of fear of Allah or for other reasons. This is the preponderant and correct view according to the majority of scholars, and it is the madhhab of Imam Ahmad, because these are common matters that cannot be repelled.