What is the ruling on rushing the transitional takbeers and movements, and does that invalidate the prayer? And what is the ruling on rushing the Fatiha and the Tashahhud while maintaining a state of tranquility equivalent to two tasbeehs?

If speeding up recitation does not impair the articulation of letters, then the recitation is sufficient, and the prayer is valid. However, if it results in a mispronunciation that alters the meaning in Al-Fatihah, then the prayer is invalid. The last Tashahhud is similar to Al-Fatihah in this regard, as it is a pillar. As for the first Tashahhud and the takbir of transition, they are Sunnah acts, and the prayer is not invalidated by mispronouncing them or by omitting them. Tranquility (طمأنينة) is the stillness and stability of the limbs for a certain period, and it is achieved simply by the worshiper remaining still during the pillar, even if for a short time.