What is the ruling on the takbir for standing for the third rak'ah and raising the hands: should it be while the Imam is sitting, or upon standing, or upon straightening up?

The Sunnah is to say the Takbir during the transition in prayer, not before or after it. The evidence for this is the agreed-upon Hadith of Abu Hurairah and Ibn Abbas. The view of the majority of scholars is that the Takbir is said during the transition, and this is the correct view. As for raising the hands, there is flexibility in the matter, and what is authentically reported from the Prophet (peace be upon him) regarding raising the hands during the opening Takbir (Takbirat al-Ihram) is raising them before, after, and concurrently with the Takbir.