Is it obligatory for one praying a fard (obligatory) prayer while sitting to say the opening takbeer (takbeerat al-ihram) while standing, and is he required to repeat the prayer if he said the takbeer while sitting?

Standing in obligatory prayers is a pillar without which the prayer is invalid. Therefore, no one is permitted to pray sitting unless they are unable to stand. The opening takbir (Takbirat al-Ihram) must be uttered with all its letters while standing. If even one letter of it is uttered while not standing, the obligatory prayer is not validly commenced. The rule for praying for the sick person is that they must perform all the duties and pillars of the prayer that they are able to, and what they are unable to perform is waived for them. So, if they are able to begin the prayer standing, they must do so, and then if standing becomes difficult for them, they may sit.