If a person praying in a chair is able to bow but unable to prostrate, is their prayer invalidated if they bow and prostrate while sitting in the chair due to not performing the form of bowing and prostration on the ground? And if they are able to stand for each rak'ah but only stand for the first rak'ah and then sit for the other rak'ahs, is their prayer invalidated due to not fulfilling the pillar of standing, and are they required to repeat it?

Standing in obligatory prayer is a duty for one who is able, and it is permissible for the incapacitated to sit. However, one who is only unable to prostrate is not permitted to pray sitting. Rather, he should pray standing, then bow, and after rising from bowing, he should sit to gesture for prostration. Therefore, if someone prays sitting while being able to bow but not prostrate, his prayer is invalid. Similarly, if someone is able to stand for every rak'ah but then sits for the other rak'ahs, his prayer is invalid. As for sitting on a chair between the two prostrations or for the tashahhud (definition) for someone unable to bend their feet, there is no harm in it due to the excuse, and invalid prayers must be repeated.