Does placing the hand on the ground to stand up from the first Tashahhud, then returning it to the thigh to repeat the Tashahhud, or descending downwards from the Ruku‘ to stand up, then pausing to say something, then standing up, count as an addition to the prayer?

What you asked about regarding standing up from the first Tashahhud, or from the Rukūʿ, is considered a slight addition that does not invalidate the prayer, nor does it necessitate the prostration of forgetfulness (Sujūd al-Sahw); because it is a minor movement. Similarly, repeating the Tashahhud does not require anything. The phrase: "If I wanted to stand up from Rukūʿ, I would go down; so that I can go up again," is not clear.