Is it permissible to place the bed facing the Qibla, such that the feet are towards it and the head is in the opposite direction, due to the room's design and the air conditioner's placement?

There is no harm in pointing one's feet towards the Qibla during sleep, as there is no prohibition against it. What is mentioned is the recommendation to sleep on one's right side. Some jurists have stated that pointing one's feet towards the Qibla is the prescribed method for a sick person to pray if they are unable to sit, which indicates that there is no religious impediment to it.