Is the recitation of Al-Fatiha alone sufficient in all rak'ahs of the prayer, and does prayer become invalidated by speed in tasbeeh during bowing and prostration, and is it obligatory to repeat the prayers that were performed in this manner?

Limiting oneself to Al-Fatiha in prayer is permissible, and reciting a surah after it is a Sunnah, the omission of which does not invalidate the prayer. Rushing in prayer, if it violates the pillar of tranquility (ṭumaʾnīnah), invalidates it. If it does not violate it, the prayer is valid. The standard for tranquility is to be settled for the duration of a single glorification (tasbīḥ). The worshiper must observe humility (khushūʿ) and tranquility, and not peck the prayer like a rooster pecks, for this resembles the action of hypocrites.