Is "tranquility" in prayer merely stillness for the purpose of separating the pillars (arkān), and does slight movement during bowing (rukūʿ) invalidate the prayer, and is prayer valid behind an imam who sways excessively during bowing?

Slight trembling of the body does not negate ṭuma’nīnah (tranquility/stillness). One should accustom themselves to stillness in prayer, due to the Prophet's (peace and blessings be upon him) saying: "Be still in prayer." Al-Nawawi said: "And in it is the command for stillness in prayer, and for devotion and attentiveness to it." Ibn Taymiyyah said: "'Be still' necessitates stillness in every part of the prayer." Minor, involuntary movements do not negate ṭuma’nīnah.