Is it obligatory for my mother to pray standing for the recitation of Al-Fatiha and then prostrate on a chair, or should she pray sitting throughout the entire prayer based on the doctor's order? And is it permissible for her to lean her back if she prays sitting?

Standing in the obligatory prayer is a pillar without which the prayer is invalid, except for an excuse. Among the excuses are: if standing causes severe hardship, or if one fears an increase in illness or a delay in recovery. This relies on the statement of a trustworthy, just, Muslim doctor. If he says that standing increases harm or delays recovery, then it is permissible to act upon his statement, and it will be an excuse for not standing. If the matter is unclear, another doctor should be consulted to confirm whether standing would cause harm or delay recovery.