Is it correct to say “Allahu Akbar” and raise the hands when standing for the third rak‘ah after the first tashahhud, and is the prayer invalidated if “Allahu Akbar” is said and the hands are raised before fully straightening up to stand?

The place for the takbirat al-intiqal (transition takbirs) and raising the hands with them is at the beginning of the transition. However, the Malikis made an exception for the takbir for standing up for the third rak'ah, saying that it is mustahabb (recommended) after the worshiper has straightened up into a standing position. What you do, raising your hands and making the takbir after standing straight for the third rak'ah, is correct according to the Malikis, and contrary to what is more preferable according to others, but your prayer is valid according to everyone.

The takbir for transition is, according to the more preponderant opinion, one of the obligatory acts of prayer. Raising the hands with it is a sunnah established from the Prophet, peace and blessings be upon him. It is makruh (disliked) to make the takbir and raise the hands before or after the transition, but the prayer is not invalidated by that.