Is it permissible for the Imam to sit when tired during prayer, and should the congregants follow him in this? Is it permissible for him to prostrate for the recitation of a verse containing a prostration during prayer, then complete the verse and bow? And is it obligatory to complete the recitation of Al-Fatiha in congregational prayer if time does not permit before the Imam begins another verse?

It is permissible for the Imam to sit if he is unable to stand during prayer, and the congregants must complete the prayer standing. This is the view of the Hanbalis. It is also permissible for the Imam to perform the prostration of recitation (Sujud al-Tilawah), and the congregants must follow him. As for reciting Al-Fatihah behind the Imam, it is more cautious for the congregant to recite and complete it, even if the Imam has started reciting a [different] surah.