Is it permissible for my father to follow in prayer the muezzin who is praying seated on a chair due to his inability, knowing that my father entered the mosque and found them both praying, and should the muezzin have put the child forward to lead the prayer, or should they have prayed individually, and if it is permissible to follow, should my father pray standing or sitting?

Scholars have differed on the validity of the Imamate (leadership in prayer) of someone who omits some pillars of prayer. The majority forbid it, while the Shafi'is permitted it, and Ibn 'Uthaymeen chose this view.

It is stated in the Fiqh Encyclopedia: "It is not permissible for someone who is able to perform a pillar... to pray behind someone who is unable to do so, according to the Malikis and Hanbalis, and it is also the view of Muhammad from the Hanafis... It is permissible for a standing person to pray behind a sitting person who performs bowing and prostration, according to Abu Hanifa and Abu Yusuf. The Shafi'is also permitted this, even if the sitting person is unable to perform bowing or prostration, based on the hadith of Aisha (may Allah be pleased with her) that the Prophet (peace be upon him) prayed the last part of his prayer sitting, while the people behind him were standing."

Therefore, there is no harm for your father in praying behind this man, nor for this man in praying behind this boy, nor in letting the boy lead the prayer if he performs it well. If someone who is unable to perform bowing and prostration leads the prayer as an Imam (according to the view of the Shafi'is and Ibn 'Uthaymeen), then the follower must perform bowing and prostration completely.

They differed on whether one should pray behind him sitting or standing. Ibn 'Uthaymeen chose the view that it is obligatory to pray behind him sitting, while the majority held that if someone prays sitting, the people pray behind him standing. As for the Hanbalis, the Imamate of a sitting person for a standing person is not valid unless he is the Imam of the mosque whose illness is expected to cease, in which case it is commendable to pray behind him sitting.

Thus, your father's prayer is valid, Allah willing, and he is not required to repeat it.