Is it permissible to address the Prophet (peace be upon him) during prayer, and what about saying "Peace be upon you, O Prophet" in the Tashahhud, is this not addressing him?

The prayer is not invalidated by directly addressing the Prophet (peace and blessings of Allah be upon him) for those who lived in his time, nor is it invalidated by addressing him in the Tashahhud. It was obligatory to respond to him if he spoke to someone, whether they were praying or not, during his era, and their prayer would not be invalidated by that according to the majority of scholars.